The stimulation of instability and transport in the bottom boundary layer by internal solitary waves has been documented for over twenty years. However, the challenge of shallow slopes and a disparity of scales between the large-scale wave and the small-scale boundary layer has proven challenging for simulations. We present laboratory scale simulations that resolve the three-dimensionalisation in the boundary layer during the entire shoaling process. We find that the late stage, in which the incoming wave fissions into boluses, provides the most consistent source of three-dimensionalisation. In the early stage of shoaling, three-dimensionalisation occurs not so much due to separation bubble instability, but due to the interaction of vortices shed from the separation bubble with the overlying pycnocline. This interaction overturns the pycnocline, and creates bursts in kinetic energy and viscous dissipation, suggesting that the shed vortices induce turbulent motion and sediment resuspension in the water column above and behind the separation bubble.

Linear unsteady aerofoil theory, while successfully used for the prediction of unsteady aerofoil lift for many decades, has yet to be proven adequate for predicting the propulsive performance of oscillating aerofoils. In this paper we test the hypothesis that the central shortcoming of linear small-amplitude models, such as the Garrick function, is the failure to account for the flow acceleration caused by aerofoil thrust. A new analytical model is developed by coupling the Garrick function to a cycle-averaged actuator disc model, in a manner analogous to the blade-element momentum theory for wind turbines and propellers. This amounts to assuming the Garrick function to be locally valid and, in combination with a global control volume analysis, enables the prediction of flow acceleration at the aerofoil. The new model is demonstrated to substantially improve the agreement with large-eddy simulations of an aerofoil in combined heave and pitch motion.

Whether colliding drops will merge with or bounce off each other is critical to numerous processes, and the physics involved is notoriously complex. In particular, experiments show that both sufficiently slow and fast head-on drop collisions lead to merging, but that there is often an intermediate regime in which bouncing is observed; these transitions in behaviour were recently discovered to be surprisingly sensitive to the radius of the drops and the ambient gas pressure. We show here that these transitions between bouncing and merging are governed by nanoscale phenomena; namely, gas-kinetic and disjoining pressure effects. To capture these crucial effects, a novel, open-source computational model is developed for the simulation of colliding drops. The model uses a hybrid approach, based on solving the Navier–Stokes equations in the drop with a lubrication approach for the unconventional physics of the gas film. Our simulations show remarkably good agreement with experiments of head-on collisions and also provide new experimentally verifiable predictions.

The first experimental results on pattern transitions in the co-rotation regime (i.e. the rotation ratio $\varOmega = \omega _o/\omega _i > 0$, where $\omega _i$ and $\omega _o$ are the angular speeds of the inner and outer cylinders, respectively) of the Taylor–Couette flow (TCF) are reported for a neutrally buoyant suspension of non-colloidal particles, up to a particle volume fraction of $\phi = 0.3$. While the stationary Taylor vortex flow (TVF) is the primary bifurcating state in dilute suspensions ($\phi \leq ~0.05$), the non-axisymmetric oscillatory states, such as the spiral vortex flow (SVF) and the ribbon (RIB), appear as primary bifurcations with increasing particle loading, with an overall de-stabilization of the primary bifurcating states (TVF/SVF/RIB) being found with increasing $\phi$ for all $\varOmega \geq ~0$. At small co-rotations ($\varOmega \sim 0$), the particles play the dual role of stabilization ($\phi < 0.1$) and destabilization ($\phi \geq ~0.1$) on the secondary/tertiary oscillatory states. The distinctive features of the ‘particle-induced’ spiral vortices are identified and contrasted with those of the ‘fluid-induced’ spirals that operate in the counter-rotation regime.

We discuss the modal, linear stability analysis of generalized Couette–Poiseuille (GCP) flow between two parallel plates moving with relative speed in the presence of an applied pressure gradient vector inclined at an angle $0\leqslant \phi \leqslant 90^\circ$ to the plate relative velocity vector. All possible GCP flows can be described by a global Reynolds number $Re$, $\phi$ and an angle $0\leqslant \theta \leqslant 90^\circ$, where $\cos \theta$ is a measure of the relative weighting of Couette flow to the composite GCP flow. This provides a novel and uncommon group of generally three-dimensional base velocity fields with wall-normal twist, for which Squire's theorem does not generally apply, requiring study of oblique perturbations with wavenumbers $(\alpha,\beta )$. With $(\theta,\phi )$ fixed, the neutral surface $f(\theta,\phi ;Re,\alpha,\beta )=0$ in $(Re, \alpha,\beta )$ space is discussed. A mapping from GCP to plane Couette–Poiseuille flow stability is found that suggests a scaling relation $Re^*\alpha /k = H(\theta ^*)$ that collapses all critical parameters, where ${Re}^*= Re\,({\alpha _1}/{\alpha })\,({\sin \theta }/{\sin {\theta }^*})$ and $\tan \theta ^*=({\alpha _{1}}/\alpha )\tan \theta$, with $\alpha _1=\alpha \cos \phi +\beta \sin \phi$. This analysis does not, however, directly reveal global critical properties for GCP flow. The global $Re_{cr}(\theta,\phi )$ shows continuous variation, while $\alpha _{cr}(\theta,\phi )$ and $\beta _{cr}(\theta,\phi )$ show complex behaviour, including discontinuities owing to jumping of critical states across neighbouring local valleys (in $Re$) or lobes of the neutral surface. The discontinuity behaviour exists for all low $\phi$. For $\phi \gtrsim 21^\circ$, variations of $\alpha _{cr}(\theta )$ and $\beta _{cr}(\theta )$ are generally smooth and monotonic.

We consider steady surface waves in an infinitely deep two-dimensional ideal fluid with potential flow, focusing on high-amplitude waves near the steepest wave with a 120$^{\circ }$ corner at the crest. The stability of these solutions with respect to coperiodic and subharmonic perturbations is studied, using new matrix-free numerical methods. We provide evidence for a plethora of conjectures on the nature of the instabilities as the steepest wave is approached, especially with regards to the self-similar recurrence of the stability spectrum near the origin of the spectral plane.

In the absence of large-scale coherent structures, a widely used statistical theory of two-dimensional turbulence developed by Kraichnan, Leith, and Batchelor (KLB) predicts a power-law scaling for the energy, $E(k)\propto k^\alpha$ with an integral exponent $\alpha ={-3}$, in the inertial range associated with the direct cascade. A power-law scaling is also observed in the presence of coherent structures, but the scaling exponent becomes fractal and often differs substantially from the value predicted by the KLB theory. Here we present a dynamical theory that sheds new light on the relationship between the spatial and temporal structure of the large-scale flow and the scaling of small-scale structures representing filamentary vorticity. Specifically, we find hyperbolic regions of the large-scale flow to play a key role in the flux of enstrophy between scales. Small-scale vorticity in these regions can be described by dynamically self-similar solutions of the Euler equation, which explains the power-law scaling. Furthermore, we find that correlations between different hyperbolic regions are responsible for the emergence of fractal scaling exponents.

Waves are formed on the surface of a sessile drop driven through substrate vibrations oriented at a slanting angle from the normal. A mathematical model is derived, which leads to an infinite system of coupled Mathieu equations governing the wave dynamics that are solved using Floquet theory. The spatial structure of the waves is described by the mode number pair $[\ell,m]$, where $\ell$ and $m$ are the polar and azimuthal mode numbers, respectively. Limiting cases corresponding to horizontal and vertical vibrations are discussed with predictions agreeing well with prior literature. We focus our results on three drop motions – (1) harmonic $[1,1]$ rocking mode, (2) harmonic $[2,0]$ pumping mode, and (3) subharmonic rocking $[1,1]$ mode – as they depend upon the slanting angle, static contact angle, and contact-line conditions, which we assume to be either pinned or freely moving with fixed contact angle. New theoretical predictions are tested through experiments over a range of parameters, showing good agreement.

During stroke reversals, insect wings interact with their own wake flow from the preceding half-stroke, resulting in an unsteady aerodynamic mechanism known as ‘wing–wake interaction’ or ‘wake capture’. To better elucidate this mechanism, we numerically solved the incompressible Navier–Stokes equations at Reynolds numbers $10^2$ and $10^3$. Simulations were conducted for wing planforms defined using the beta function distribution with varying aspect ratios ($AR=2\unicode{x2013}6$) and radial centroid locations ($\hat {r}_1=0.4\unicode{x2013}0.6$), whilst employing representative normal hovering kinematics. The wake development from the considered flapping wing planforms was investigated, and the wake capture contribution to aerodynamic force production was quantified by comparing the force generation between the fifth and first stroke cycles at multiple sections along the wingspan. Our results revealed that on the inboard wing region experiencing an attached leading-edge vortex (LEV) structure, wing–wake interaction is dominated by an unsteady downwash effect, resulting in a reduction in local force production. However, in regions closer to the wingtip experiencing detachment of the LEV, wing–wake interaction is dominated by an unsteady upwash effect, leading to an increase in local force production. Consequently, the global wake capture force production is controlled by the extent of LEV detachment, which primarily increases with the increase of wing aspect ratio. This suggests that for normal hovering flapping wings, the typical loss in translational force production due to wingtip stall is partially mitigated by wake capture effects.

The asymptotic analysis of steady azimuthally invariant electromagnetically driven flows occurring in a shallow annular layer of electrolyte undertaken in Part 1 of this study (McCloughan & Suslov, J. Fluid Mech., vol. 980, 2024, A59) predicted the existence of a two-tori flow state that has not been detected previously. In Part 2 of the study we confirm its existence by numerical time integration of the governing equations. We observe a hysteresis, where the type of solution obtained for the same set of governing parameters depends on the choice of the initial conditions and the way the governing parameters change, which is fully consistent with the analytic results of Part 1. Subsequently, we perform a linear stability analysis of the newly obtained steady state and deduce that the experimentally observed anti-cyclonic free-surface vortices appear on its background as a result of a centrifugal (Rayleigh-type) instability of the interface separating two counter-rotating toroidal structures that form the newly found flow solution. The quantitative characteristics of such instability structures are determined. It is shown that such structures can only exist in sufficiently thin layers with the depth not exceeding a certain critical value.

We explored the instability dynamics of the viscous fingering interaction in dual displacement fronts by varying the viscosity configuration. Four regimes of rear-dominated fingering, front-dominated fingering, dual fingering and stable were identified. By using the breakthrough time, which refers to the breakup of the dual displacement fronts, the instability dynamics were modelled, and a regime map was developed. These serve as a tool for effectively harnessing the dual displacement fronts for various applications, such as hydrogeology, petroleum, chemical processes and microfluidics.

Acoustic resonance is an important factor that contributes to aeroengine compressor failure. In this study, a plane cascade of compressor blades was designed to reproduce acoustic resonance via a low-speed wind tunnel test. A high-frequency hot-wire, microphone and strain gauge were used to synchronously measure the fluid, acoustic and structural parameters. We analysed the variation in the amplitude and frequency of the multi-field parameters with increasing mean flow velocity and explored the multi-field interaction mechanism that induces the acoustic resonance of the plane cascade. The plane cascade effectively reproduced the acoustic resonance phenomenon. The first-order acoustic-mode frequency of the plane cascade flow duct, second-order torsional vibration mode frequency of the blade and shedding mode frequency of the tip clearance leakage vortex were equal under acoustic resonance. The fluid, acoustic and structural fields showed a strong interaction effect, achieving the maximum blade vibration amplitude and causing fatigue cracks of torsional vibration at the blade root. The frequency lock-in region of the compressor plane cascade was divided into an ‘acoustic–structure’ interaction region, a ‘fluid–acoustic–structure’ interaction region and a first-order acoustic-mode dominant region with increasing mean flow velocity, which demonstrates an interesting phenomenon in which the fluid–acoustic–structure modes compete: acoustic mode > blade vibration mode > vortex shedding mode. The results demonstrate a unique approach to the study of acoustic resonance that provides insight into the acoustic resonance mechanism in a cascade of compressor blades.

We investigate air-entraining flows where degassing, rather than fragmentation, plays a significant role. Of interest is the power-law slope $\beta$ of the bulk bubble size distribution $N(a)$ during the air-generating period, when the total volume of bubbles is increasing. We study a canonical air-entraining flow created by strong underlying free-surface turbulence. We perform analysis using the population balance equation (PBE) and computations using direct numerical simulations (DNS) with bubble tracking. We quantify the importance of degassing by the ratio of degassing flux ($Q_D$) to entrainment flux ($Q_I$), $\mathcal {D}=Q_D/Q_I$, and the ratio of degassing rate ($\varLambda (a)$) to fragmentation rate ($\varOmega (a)$) for a bubble of radius $a$, $\varLambda (a)/\varOmega (a)$. For a broad range of large Froude numbers ${{Fr}}=U/\sqrt {L g}$, DNS give $\mathcal {D}=\operatorname {O}(1)$ (independent of ${{Fr}}$), showing that degassing is relevant, and $\varLambda (a) \gg \varOmega (a)$, showing that the bubble population is degassing-dominated. In contrast to fragmentation-dominated populations, such as those due to wave breaking, where $\beta =-10/3$, degassing-dominated populations have qualitatively different $N(a)$ during air entrainment. Analysis using the PBE shows that degassing-dominated $\beta$ is a function of $\varLambda (a)$, which has a turbulence-driven regime ($a< a_\varLambda$) and a buoyancy-driven regime ($a>a_\varLambda$). Here, $a_\varLambda$ is the bubble radius where terminal buoyant rise velocity equals $u_{rms}$. Consequently, $N(a)$ exhibits a split power with $\beta (a< a_\varLambda )=-4.\bar {3}$ and $\beta (a>a_\varLambda )=-5.8\bar {3}$ for moderate bubble Reynolds numbers ${{Re}}_b$. For large ${{Re}}_b$, $\beta (a>a_\varLambda )=-4.8\bar {3}$. The DNS strongly confirm these findings for moderate ${{Re}}_b$. By identifying and describing degassing-dominated bubble populations, this work contributes to the understanding and interpretation of broad types of air-entraining problems where degassing plays a relevant role.

We use three-dimensional direct numerical simulations of homogeneous isotropic turbulence in a cubic domain to investigate the dynamics of heavy, chiral, finite-size inertial particles and their effects on the flow. Using an immersed-boundary method and a complex collision model, four-way coupled simulations have been performed, and the effects of particle-to-fluid density ratio, turbulence strength and particle volume fraction have been analysed. We find that freely falling particles on the one hand add energy to the turbulent flow but, on the other hand, they also enhance the flow dissipation: depending on the combination of flow parameters, the former or the latter mechanism prevails, thus yielding enhanced or weakened turbulence. Furthermore, particle chirality entails a preferential angular velocity which induces a net vorticity in the fluid phase. As turbulence strengthens, the energy introduced by the falling particles becomes less relevant and stronger velocity fluctuations alter the solid phase dynamics, making the effect of chirality irrelevant for the large-scale features of the flow. Moreover, comparing the time history of collision events for chiral particles and spheres (at the same volume fraction) suggests that the former tend to entangle, in contrast to the latter which rebound impulsively.

The study investigates the flow structure, dynamics of the large-scale coherent structures, drag forces and sediment entrainment mechanisms generated by a circular array of diameter D containing rigid emerged circular cylinders of diameter d placed in a smooth-bed channel of depth h under strong shallow flow conditions (D/h = 20, d/D = 0.0125 and 0.025). Eddy resolving simulations are conducted with different values of the solid volume fraction 0.025 ≤ SVF ≤ 0.1 and non-dimensional frontal area per unit volume (2.56 ≤ aD ≤ 10.2, where for the present configuration, a = SVF(1/d)4/${\rm \pi}$) and a fixed channel Reynolds number (Reh = 10 000). The flow conditions are such that a vortex street (VS)-type of shallow wake is expected to form for a solid cylinder of diameter D. Findings are compared with previous results obtained for cases with moderately shallow conditions 2.5 ≤ D/h ≤ 3.5 and with the limiting case of flow past a solid cylinder with D/h = 20. For moderately shallow conditions, the core of the main horseshoe vortex (HV) forming around the array occupies a small fraction of the water column and its coherence is the largest in front of the array. By contrast, for very shallow conditions, multiple HVs form around the array and their cores occupy a large fraction of the water column. Moreover, the coherence of most of these HVs peaks close to the sides of the array. Only for relatively large aD values, the main HV extends over the whole upstream face of the array, similar to the limiting case of a solid cylinder. For sufficiently high aD, a secondary instability is present inside the near wake that leads to the formation of parallel horizontal vortices in the vicinity of the wake roller vortices. The changes in the wake structure with decreasing SVF and aD are qualitatively similar to those observed for cases with moderately shallow flow conditions, with the antisymmetric wake shedding mode being suppressed for aD ≤ 2.5. However, the Strouhal number associated with the shedding of wake rollers, which is still close to 0.2 for a solid cylinder with D/h = 20, can be as high as 0.4 for aD = 2.5. The paper also discusses how the steady wake and total wake lengths and the strength of the bleeding flow vary with the SVF. Simulation results show that the capacity of the flow to entrain sediment inside and around the array peaks for SVF = 0.05 and aD = 5.2, with sediment entrainment monotonically increasing with the SVF outside the array and monotonically decreasing inside the array. Despite the differences in the flow structure next to and inside the array, the variation of the mean, time-averaged streamwise drag coefficient of the solid cylinders ${\bar{C}_d}$ with aD is close to that observed for arrays with moderately shallow flow conditions. The combined drag coefficient for the array decreases with increasing flow shallowness, with the decay being stronger for relatively large values of aD.

The entrainment of ambient fluid into a variable-density jet is typically quantified using an entrainment coefficient $\alpha$. Here, we investigate the dependence of $\alpha$ on the ratio of the jet's density $\rho _m$ and that of the ambient fluid $\rho _0$. Current parametrisations of $\alpha$ rely on a scaling inferred from early laboratory experiments (Ricou & Spalding, J. Fluid Mech., vol. 11, 1961, pp. 21–32). We demonstrate analytically that the experiments preclude definitive conclusions regarding the dependence of $\alpha$ on $\rho _m / \rho _0$ and that the underlying physical processes therefore warrant closer attention. To investigate the physics behind the dependence of entrainment on the density ratio we use a Favre-averaged entrainment decomposition. The decomposition is applied to data from large-eddy simulations of jets characterised by density ratios $\rho _m / \rho _0$ spanning over two orders of magnitude that have been verified against experimental data. Changes in the shape of the velocity profile are a significant contributor to entrainment in the near field due to the breakdown of the potential core, and persist over larger streamwise distances in heavy releases than in light releases. Therefore, to focus exclusively on the effects of density ratio, we study the region where the shape changes have become small but the density ratio is still significant. We show that the dimensionless turbulent kinetic energy production and mean kinetic energy flux depend strongly on the density ratio, both for our large-eddy simulation data and for recent experiments. Despite this, the entrainment coefficient is practically constant in this region and has value $\alpha \approx 0.07$ for all simulations.

We perform direct numerical simulations of actively controlled laminar separated wakes around low-aspect-ratio wings with two primary goals: (i) reducing the size of the separation bubble and (ii) attenuating the wing tip vortex. Instead of preventing separation, we modify the three-dimensional (3-D) dynamics to exploit wake vortices for aerodynamic enhancements. A direct wake modification is considered using optimal harmonic forcing modes from triglobal resolvent analysis. For this study, we consider wings at angles of attack of $14^\circ$ and $22^\circ$, taper ratios $0.27$ and $1$, and leading edge sweep angles of $0^\circ$ and $30^\circ$, at a mean-chord-based Reynolds number of $600$. The wakes behind these wings exhibit 3-D reversed-flow bubble and large-scale vortical structures. For tapered swept wings, the diversity of wake vortices increases substantially, posing a challenge for flow control. To achieve the first control objective for an untapered unswept wing, root-based actuation at the shedding frequency is introduced to reduce the reversed-flow bubble size by taking advantage of the wake vortices to significantly enhance the aerodynamic performance of the wing. For both untapered and tapered swept wings, root-based actuation modifies the stalled flow, reduces the reversed-flow region and enhances aerodynamic performance by increasing the root contribution to lift. For the goal of controlling the tip vortex, we demonstrate the effectiveness of actuation with high-frequency perturbations near the tip. This study shows how insights from resolvent analysis for unsteady actuation can enable global modification of 3-D separated wakes and achieve improved aerodynamics of wings.

As shown by Wenzel et al. (J. Fluid Mech., vol. 930, 2022, A1), the Eckert number $Ec$ defined using the difference between recovery temperature $\bar{T}_r$ and wall temperature $\bar{T}_w$ can be understood as a meaningful quantity to compare heat-transfer effects inside compressible turbulent boundary layers (for a calorically perfect gas), no matter whether these are caused by different Mach-number or wall-temperature conditions. While the named study deduced this comparative behaviour of $Ec$ from an integral perspective in a strict sense, Cogo et al. (J. Fluid Mech., vol. 974, 2023, A10) performed a systematic parameter study based on the previous findings to look at wall-normal profiles. They have shown that the diabatic parameter $\varTheta$, being equivalent to $Ec$, is capable of categorizing heat-transfer effects for cases at different Mach numbers, even to some extent for some of the wall-normal profiles. Building on this progress, the present paper provides a comprehensive classification of both existing and newly computed super- and hypersonic direct numerical simulation data at various wall temperature conditions into heated cases, adiabatic cases or weakly/moderately/strongly/quasi-incompressibly cooled cases. Hereby, the classification is largely based on the wall-normal position of the temperature peak occurring in cooled boundary-layer cases, which is one of the determining factors for the topological characteristics of diabatic boundary-layer profiles. Integrating high-enthalpy data into the analysis allowed us to confirm the reliability of the proposed classification also in more complex scenarios, where the calorically perfect gas assumption no longer applies and additional heat-transfer mechanisms come into play. While the Eckert number is shown to well characterize heat-transfer effects on most important temperature-related quantities for a wide range of Mach numbers and $\bar {T}_w/\bar {T}_r$ conditions, also the local Reynolds number $Re_{\tau }$ is shown to notably affect the strength of heat-transfer effects. Since both $Ec$ and $Re_{\tau }$ can be determined in advance – or estimated to a reasonable extent – a key advantage of the classification scheme is to allow for an effective a priori estimation of the extent to which heat-transfer effects are to be expected for a given compressible turbulent boundary-layer configuration.

We study the dynamics of hydraulic fracturing of an elastic solid in a Hele-Shaw cell. Compared with hydraulic fractures in an infinite elastic bulk, the viscous resistance comes mainly from the drag by the two parallel plates that forms the Hele-Shaw cell rather than by the fluid–solid interface. Such a feature leads to a different nonlinear differential–integral system that describes the coupled evolution of the fracture shape and pressure field. Our theory leads to hydraulic fractures of cusp shapes in the neighbourhood of the fracture tip, which is consistent with recent experimental observations. Accordingly, there exists no pressure singularity at the location of the fracture tip, which is also fundamentally different from our previous understandings of hydraulic fracturing of elastic solids.

Communities of swimming microorganisms often thrive near liquid–air interfaces. We study how such ‘active carpets’ shape their aquatic environment by driving biogenic transport in the water column beneath them. The hydrodynamic stirring that active carpets generate leads to diffusive upward fluxes of nutrients from deeper water layers, and downward fluxes of oxygen and carbon. Combining analytical theory and simulations, we examine the biogenic transport by studying fundamental metrics, including the single and pair diffusivity, the first passage time for particle pair encounters and the rate of particle aggregation. Our findings reveal that the hydrodynamic fluctuations driven by active carpets have a region of influence that reaches orders of magnitude further in distance than the size of the organisms. These non-equilibrium fluctuations lead to a strongly enhanced diffusion of particles, which is anisotropic and space dependent. Fluctuations also facilitate encounters of particle pairs, which we quantify by analysing their velocity pair correlation functions as a function of distance between the particles. We found that the size of the particles plays a crucial role in their encounter rates, with larger particles situated near the active carpet being more favourable for aggregation. Overall, this research broadens our comprehension of aquatic systems out of equilibrium and how biologically driven fluctuations contribute to the transport of fundamental elements in biogeochemical cycles.