The linear instability of circular vortices over isolated topography in a homogeneous and inviscid fluid is examined for the shallow-water and quasi-geostrophic models in the $f$-plane. The eigenvalue problem associated with azimuthal disturbances is derived for arbitrary axisymmetric topographies, either submarine mountains or valleys. Amended Rayleigh and Fjørtoft theorems with topographic effects are given for barotropic instability, obtaining necessary criteria for instability when the potential vorticity gradient is zero somewhere in the domain. The onset of centrifugal instability is also discussed by deriving the Rayleigh circulation theorem with topography. The barotropic instability theorems are applied to a wide family of nonlinear, quasi-geostrophic solutions of circular vortices over axisymmetric topographic features. Flow instability depends mainly on the vortex/topography configuration, as well as on the vortex size in comparison with the width of the topography. It is found that anticyclones/mountains and cyclones/valleys may be unstable. In contrast, cyclone/mountain and anticyclone/valley configurations are stable. These statements are validated with two numerical methods. First, the generalised eigenvalue problem is solved to obtain the wavenumber of the fastest-growing perturbations. Second, the evolution of the vortices is simulated numerically to detect the development of linear perturbations. The numerical results show that for unstable vortices over narrow topographies, the fastest growth rate corresponds to mode $1$, which subsequently forms asymmetric dipolar structures. Over wide topographies, the fastest perturbations are mainly modes $1$ and $2$, depending on the topographic features.

Biologically inspired finlet treatments have been shown to effectively reduce the trailing-edge noise of a flat plate and hence are a viable noise-suppression technology for engineering applications. The present work performs a thorough experimental investigation on the near-field dynamics of finlet surface treatments applied to a flat plate. To examine the underlying noise-reduction mechanism, the manipulated flow field is analysed using data from detailed static, unsteady wall-pressure as well as velocity measurements and their correlations. Specifically, the densely populated dynamic transducers allow for the tracking of the turbulent boundary-layer development from upstream to the wake of the finlet-treated area (see supplementary movies), which elucidates the formation of ‘finlet-induced turbulence’ through flow–finlet interaction. Associated turbulence structures are found to further develop within the treated area and structures shed from the top of the finlets are observed to mix and merge with the turbulence being channelled through the space between the finlets in the finlet wake. While the mixing process increases the spanwise turbulence length scale, it significantly attenuates the unsteady wall-pressure fluctuation at the trailing edge and thus leads to broadband reduction of the trailing-edge noise. Moreover, it corroborates the findings of earlier studies suggesting that there exists an optimal distance between finlets and trailing-edge where the mixing effects are most beneficial.

We numerically investigate the hydrodynamics of a spherical swimmer carrying a rigid cargo in a Newtonian fluid. This swimmer model, a ‘squirmer’, which is self-propelled by generating tangential surface waves, is simulated by a direct-forcing fictitious domain method (DF-FDM). We consider the effects of swimming Reynolds numbers (Re) (based on the radius and the swimming speed of the squirmers), the assembly models (related to the cargo shapes, the relative distances (ds) and positions between the squirmer and the cargo) on the assembly's locomotion. We find that the ‘pusher-cargo’ (pusher behind the cargo) model swims significantly faster than the remaining three models at the finite Re adopted in this study; the term ‘pusher’ indicates that the object is propelled from the rear, as opposed to ‘puller’, from the front. Both the ‘pusher-cargo’ and ‘cargo-pusher’ (pusher in front of the cargo) assemblies with an oblate cargo swim faster than the corresponding assemblies with a spherical or prolate cargo. In addition, the pusher-cargo model is significantly more efficient than the other models, and a larger ds yields a smaller carrying hydrodynamic efficiency η for the pusher-cargo model, but a greater η for the cargo-pusher model. We also illustrate the assembly swimming stability, finding that the ‘puller-cargo’ (puller behind the cargo) model is stable more than the ‘cargo-puller’ (puller in front of the cargo) model, and the assembly with a larger ds yields more unstable swimming.

Explosive dispersal of granular media widely occurs in nature across various length scales, also enabling engineering applications ranging from commercial or military explosive systems to the loss prevention industry. However, the complex particle–flow coupling makes the explosive dispersal behaviour of particles difficult to control or even characterize. Here, we study the central explosion-driven dispersal of dense particle layers using the coarse-grained computational fluid dynamics–discrete element method and present a comprehensive investigation of both macroscale dispersal behaviours and particle-scale pattern formation. Employing three independent dimensionless parameters that characterize the efficiency, homogeneity and completeness of explosive dispersal, we categorize the dispersal behaviours into ideal, partial, retarded and failed modes, and propose the corresponding thresholds. As the mass ratio of granular materials to central pressurized gases (M/C) spans four orders of magnitude, the dispersal mode transitions from ideal to partial, then to retarded and finally to failed mode. The transitions of dispersal modes correspond to the particle–flow coupling regime crossovers, which change from decoupling to weak, medium and finally to strong coupling as the dispersal mode undergoes corresponding transitions. We proceed to develop continuum models accounting for the shock compaction and the ensuing pulsation of the particle ring that are capable of identifying the ideal dispersal mode from various dispersal systems. We also provide insights into the origins of diverse particle-scale patterns that are strongly correlated with macroscale dispersal modes and critical for the accurate prediction of dispersal modes.

In this work, we report numerical results on the flow instability and bifurcation of a viscoelastic fluid in the upstream region of a cylinder in a confined narrow channel. Two-dimensional direct numerical simulations based on the FENE-P model (the finite-extensible nonlinear elastic model with the Peterlin closure) are conducted with numerical stabilization techniques. Our results show that the macroscopic viscoelastic constitutive relation can capture the viscoelastic upstream instability reported in previous experiments for low-Reynolds-number flows. The numerical simulations reveal that the non-dimensional recirculation length (LD) is affected by the cylinder blockage ratio (BR), the Weissenberg number (Wi), the viscosity ratio (β) and the maximum polymer extension (L). Close to the onset of upstream recirculation, LD with Wi satisfy Landau-type quartic potential under certain parameter space. The bifurcation may exhibit subcritical behaviour depending on the values of L2 and β. The parameters β and L2 have nonlinear influence on the upstream recirculation length. This work contributes to our theoretical understanding of this new instability mechanism in viscoelastic wake flows.

When studying instability of weakly non-parallel flows, it is often desirable to convert temporal growth rates of unstable modes, which can readily be computed, to physically more relevant spatial growth rates. This has been performed using the well-known Gaster's transformation for primary instability and Herbert's transformation for the secondary instability of a saturated primary mode. The issue of temporal–spatial transformation is revisited in the present paper to clarify/rectify the ambiguity/misunderstanding that appears to exist in the literature. A temporal mode and its spatial counterpart may be related by sharing either the real frequency or wavenumber, and the respective transformations between their growth rates are obtained by a simpler consistent derivation than the original one. These transformations, which consist of first- and second-order versions, are valid under conditions less restrictive than those for Gaster's and Herbert's transformations, and reduce to the latter under additional conditions, which are not always satisfied in practice. The transformations are applied to inviscid Rayleigh instability of a mixing layer and a jet, secondary instability of a streaky flow as well as general detuned secondary instability (including subharmonic and fundamental resonances) of primary Mack modes in a supersonic boundary layer. Comparison of the transformed growth rates with the directly calculated spatial growth rates shows that the transformations derived in this paper outperform Gaster's and Herbert's transformations consistently. The first-order transformation is accurate when the growth rates are small or moderate, while the second-order transformations are sufficiently accurate across the entire instability bands, and thus stand as a useful tool for obtaining spatial instability characteristics via temporal stability analysis.

In the propagation and evolution of sea waves, previous studies pointed out that the occurrence of the freak wave height is significantly related to the quasi-resonant four-wave interaction in the modulated waves. From numerical--experimental study over an uneven bottom, the nonlinear effect caused by the bathymetry change also contributes to the occurrence of extreme events in unidirectional waves. To comprehensively analyse the two-dimensional wavefield, this study develops an evolution model for a directional random wavefield based on the depth-modified nonlinear Schrödinger equation, which considers the nonlinear resonant interactions and the wave shoaling the shallow water. Through Monte Carlo simulation, we discuss the directional effect on the four-wave interaction in the wave train and the maximum wave height distribution from deep to shallow water with a slow varying slope. The numerical result indicates that the directional spreading has a dispersion effect on the freak wave height. In a shallow-water environment, this effect becomes weak, and the bottom topography change is the main influencing factor in the wave evolution.

The events following the 15 January 2022 explosions of the Hunga Tonga-Hunga Ha'apai volcano highlighted the need for a better understanding of ocean-atmosphere interactions when large amounts of energy are locally injected into one (or both). Starting from the compressible Euler equations, a two-way coupled (TWC) system is derived governing the long-wave behaviour of the ocean and atmosphere under isentropic constraint. Bathymetry and topography are accounted for along with three-dimensional atmospheric non-uniformities through their depth average over a spherical shell. A linear analysis, yielding two pairs of gravito-acoustic waves, offers explanations for phenomena observed during the Tonga event. A continuous transcritical regime (in terms of water depth) is identified as the source of large wave generation in deep water bodies, removing the singularity-driven Proudman-type resonance observed in one-way coupled models. The refractive properties, governing the interaction of the atmospheric wave with step changes in water depth, are derived to comment on mode-to-mode energy transfer. Two-dimensional global simulations modelling the propagation of the atmospheric wave (under realistic conditions on the day) and its worldwide effect on oceans are presented. Local maxima of water-height disturbance in the farfield from the volcano, linked to the atmospheric wave deformation (in agreement with observations), are identified, emphasising the importance of the TWC model for any daylong predictions. The proposed framework can be extended to include additional layers and physics, e.g. ocean and atmosphere stratification. With the aim of contributing to warning system improvement, the code necessary to simulate the event with the proposed model is made available.

The oscillatory Kelvin–Helmholtz (K–H) instability of a planar liquid sheet was experimentally investigated in the presence of an axial oscillating gas flow. An experimental system was initiated to study the oscillatory K–H instability. The surface wave growth rates were measured and compared with theoretical results obtained using the authors’ early linear method. Furthermore, in a larger parameter range experimentally studied, it is interesting that there are four different unstable modes: first disordered mode (FDM), second disordered mode (SDM), K–H harmonic unstable mode (KHH) and K–H subharmonic unstable mode (KHS). These unstable modes are determined by the oscillating amplitude, oscillating frequency and liquid inertia force. The frequencies of KHH are equal to the oscillating frequency; the frequency of KHS equals half the oscillating frequency, while the frequencies of FDM and SDM are irregular. By considering the mechanism of instability, the instability regime maps on the relative Weber number versus liquid Weber number (Werel–Wel) and the Weber number ratio versus the oscillating frequency (Werel/Wel–$\varOmega$s2) were plotted. Among these four modes, KHS is the most unexpected: the frequency of this mode is not equal to the oscillating frequency, but the surface wave can also couple with the oscillating gas flow. Linear instability theory was applied to divide the parameter range between the different unstable modes. According to linear instability theory, K–H and parametric unstable regions both exist. However, note that all four modes (KHH, KHS, FDM and SDM) corresponded primarily to the K–H unstable region obtained from the theoretical analysis. Nevertheless, the parametric unstable mode was also observed when the oscillating frequency and amplitude were relatively low, and the liquid inertia force was relatively high. The surface wave amplitude was small but regular, and the evolution of this wave was similar to that of Faraday waves. The wave oscillating frequency was half that of the surface wave.

A closed expression for the average pressure difference (often called the macroscopic dynamic capillary pressure in the literature) is proposed for two-phase, Newtonian, incompressible, isothermal and creeping flow in homogeneous porous media. This upscaled equation complements the average equations for mass and momentum transport derived in a previous article. Consistently with this work, the expression is derived employing a simplified version of the volume-averaging method that makes use of elements of the adjoint method and Green's formula. The resulting equation for the average pressure difference is novel, as it shows that this quantity is controlled by the pressure gradient (and body forces) in each phase, as well as interfacial effects, and is applicable to situations in which the fluid–fluid interface is not necessarily at its steady position. The effective-medium quantities associated with the sources are all obtained from the solution of a single adjoint (or closure) problem to be solved on a (periodic) unit cell representative of the process. The average pressure difference predicted by the derived expression is validated through excellent comparisons with direct numerical simulations performed in a model porous structure.

Collision of two counterflowing gravity currents of equal densities and heights was investigated by means of three-dimensional high-resolution simulations with the goal of understanding the flow structures and energetics in the collision region in more detail. The lifetime of collision is approximately $3 \tilde {H}/\tilde {u}_f$, where $\tilde {H}$ is the depth of heavy and ambient fluids, and $\tilde {u}_f$ is the front velocity of the approaching gravity currents, and the lifetime of collision can be divided into three phases. During Phase I, $-0.2 \leqslant (\tilde {t}-\tilde {t}_c) \tilde {u}_f/\tilde {H} \leqslant 0.5$, where $\tilde {t}$ is the time, and $\tilde {t}_c$ is the time instance at which the two colliding gravity currents have fully osculated, geometric distortions of the gravity current fronts result in stretching of pre-existing vorticity in the wall-normal direction inside the fronts, and an array of vertical vortices extending throughout the updraught fluid column develop along the interface separating the two colliding gravity currents. The array of vertical vortices is responsible for the mixing between the heavy fluids of the two colliding gravity currents and for the production of turbulent kinetic energy in the collision region. The presence of the top boundary deflects the updraughts into the horizontal direction, and a number of horizontal streamwise vortices are generated close to the top boundary. During Phase II, $0.5 \leqslant (\tilde {t}-\tilde {t}_c) \tilde {u}_f/\tilde {H} \leqslant 1.2$, the horizontal streamwise vortices close to the top boundary induce turbulent buoyancy flux and break up into smaller structures. While the production of turbulent kinetic energy weakens, the rate of transfer of energy to turbulent flow due to turbulent buoyancy flux reaches its maximum and becomes the primary supply in the turbulent kinetic energy in Phase II. During Phase III, $1.2 \leqslant (\tilde {t}-\tilde {t}_c) \tilde {u}_f/\tilde {H} \leqslant 2.8$, the collided fluid slumps away from the collision region, while the production of turbulent kinetic energy, turbulent buoyancy flux and dissipation of energy attenuate. From the point of view of energetics, the production of turbulent kinetic energy and turbulent buoyancy flux transfers energy away from the mean flow to the turbulent flow during the collision. Our study complements previous experimental investigations on the collision of gravity currents in that the flow structures, spatial distribution and temporal evolution of the mean flow and turbulent flow characteristics in the collision region are presented clearly. It is our understanding that such complete information on the energy budgets in the collision region can be difficult to attain in laboratory experiments.

The effects of perturbation-based active flow control on supersonic rectangular twin jets (SRTJ) over a wide range of nozzle pressure ratios (NPR = 2.77 to 6.7, corresponding to fully expanded Mach numbers Mj = 1.3 to 1.9) were investigated. The aspect ratio and design Mach number for the bi-conic, converging-diverging nozzles were 2 and 1.5, respectively. The flow and acoustic fields of SRTJ are known to couple, often generating high near-field (NF) pressure fluctuations and elevated far-field (FF) noise levels. Large-scale structures (LSS), or equivalently instability waves or wave packets, are responsible for mixing noise, broadband shock-associated noise, screech and coupling. The primary objective of this research was to manipulate the development of LSS in this complex flow to better understand and mitigate their effects. The organization and passage frequency of the LSS were altered by excitation of instabilities over a wide range of frequencies and modes. Key findings include: (1) the screech mode of each jet was flapping along its minor axis; (2) the jets coupled, out-of-phase primarily in overexpanded cases and in-phase primarily in underexpanded cases, along the minor axis of the SRTJ; (3) coupling has significant effects on the NF pressure fluctuations, but only minor effect on the FF noise; (4) standing waves were observed only on the minor axis plane of the SRTJ; (5) altering or suppressing coupling can significantly reduce NF pressure fluctuations; (6) two high-frequency excitation methods proved effective in reducing the FF noise; and (7) nonlinear interactions between the screech tones and excitation input were observed in controlled cases in which screech was only partially suppressed.

In this paper, the reflection of curved shock waves over a symmetry plane in planar supersonic flow is studied. This includes stable Mach reflection (MR) and the regular reflection (RR) to MR transition process. Curved shock theory (CST) is applied to derive the high-order parameters in front of and behind the shock wave. The method of curved shock characteristics is used to establish an analytical model to predict the wave configurations. The shock structures provided by the proposed model agree well with the numerical results. Flow structures, such as the height of the Mach stem and the shape of the shock wave and slip line, are studied by applying the analytical model. Isentropic waves generated from a curved wall are found to significantly influence the flow patterns. It appears that the compression waves obstruct the formation of the sonic throat and increase the Mach-stem height. The expansion waves have the opposite effect. The evolution mechanism of the Mach stem is found in conjunction with the RR-to-MR transition process. The CST is extended to a moving frame and used to model the transition. The time history of the moving triple point illustrates the effects of the incident shock angle and isentropic waves on the transition process.

We derive interface models for three-dimensional Rayleigh–Taylor instability (RTI), making use of a novel asymptotic expansion in the non-locality of the fluid flow. These interface models are derived for the purpose of studying universal features associated with RTI such as the Froude number in single-mode RTI, the predicted quadratic growth of the interface amplitude under multi-mode random perturbations, the optimal (viscous) mixing rates induced by the RTI and the self-similarity of horizontally averaged density profiles and the remarkable stabilization of the mixing layer growth rate which arises for the three-fluid two-interface heavy–light–heavy configuration, in which the addition of a third fluid bulk slows the growth of the mixing layer to a linear rate. Our interface models can capture the formation of small-scale structures induced by severe interface roll-up, reproduce experimental data in a number of different regimes and study the effects of multiple interface interactions even as the interface separation distance becomes exceedingly small. Compared with traditional numerical schemes used to study such phenomena, our models provide a computational speed-up of at least two orders of magnitude.

In the fully rough regime, proposed models predict a scaling for a roughness heat-transfer coefficient, e.g. the roughness Stanton number ${St}_k \sim (k^+)^{-p} {Pr}^{-m}$ where the exponent values $p$ and $m$ are model dependent, giving diverse predictions. Here, $k^+$ is the roughness Reynolds number and ${Pr}$ is the Prandtl number. To clarify this ambiguity, we conduct direct numerical simulations of forced convection over a three-dimensional sinusoidal surface spanning $k^+ = 5.5$–$111$ for Prandtl numbers ${Pr} = 0.5$, 1.0 and 2.0. These unprecedented parameter ranges are reached by employing minimal channels, which resolve the roughness sublayer at an affordable cost. We focus on the fully rough phenomenologies, which fall into two groups: $p=1/2$ (Owen & Thomson, J. Fluid Mech., vol. 15, issue 3, 1963, pp. 321–334; Yaglom & Kader, J. Fluid Mech., vol. 62, issue 3, 1974, pp. 601–623) and $p=1/4$ (Brutsaert, Water Resour. Res., vol. 11, issue 4, 1975b, pp. 543–550). Although we find the mean heat transfer favours the $p=1/4$ scaling, the Prandtl–Blasius boundary-layer ideas associated with the Reynolds–Chilton–Colburn analogy that underpin the $p=1/2$ can remain an apt description of the flow locally in regions exposed to high shear. Sheltered regions, meanwhile, violate this behaviour and are instead dominated by reversed flow, where no clear correlation between heat and momentum transfer is evident. The overall picture of fully rough heat transfer is then not encapsulated by one singular mechanism or phenomenology, but rather an ensemble of different behaviours locally. The implications of the approach to a Reynolds-analogy-like behaviour locally on bulk measures of the Nusselt and Stanton numbers are also examined, with evidence pointing to the onset of a regime transition at even-higher Reynolds numbers.

In the present study, the shape of a two-dimensional cylinder is optimised to minimise the mean drag in laminar unsteady flow under a noisy environment. A small inline stochastic oscillation in the free-stream velocity, which follows the Ornstein–Uhlenbeck process, is considered for the noise. The small noise is found to yield a large random fluctuation in instantaneous drag of the cylinder due to the effect of added mass. Subject to the strong random fluctuation of drag, the shape optimisation is performed using an ensemble-variation-based method (EnVar), as the conventional adjoint-based optimisation is not applicable to such a flow environment with unknown free-stream noise. The optimised cylinder geometry is found to be a nearly-symmetric slender oval at a low Reynolds number. As the Reynolds number is increased, two optimal shapes emerge: one is identical to the oval obtained at the low Reynolds number, and the other is an asymmetric oval, the rear side of which is more slender than the front side, reminiscent of an aerofoil. Despite the large random fluctuation in the instantaneous drag, the optimal cylinder shapes obtained for different levels of the upstream noise are found to be almost identical. It is shown that the robust nature of the optimal cylinder shape originates from the limited influence of the small upstream noise on the mean flow properties of the cylinder wake. Finally, the optimised cylinder primarily reduces the pressure component of the drag, associated mainly with vortex shedding in the wake, and this is achieved by marginally increasing the viscous drag through the shape change.