We revisit Brenner's seminal work on the Stokes resistance of a slightly deformed sphere (Chem. Engng Sci., vol. 19, 1964, p. 519), evaluate its range of validity and extend its applicability to higher deformations for axisymmetric particles, using hydrodynamic radius as the measure of Stokes resistance. Brenner's method solves the flow around a slightly deformed sphere through two mapping steps: the first mapping translates the surface velocity on the deformed sphere to that over a reference sphere of arbitrary radius using an asymptotic expansion of the flow field in terms of deformation amplitude and a Taylor expansion of the velocity field around the surface of the reference sphere. Subsequently, the second mapping extrapolates the velocity field from the surface of the reference sphere to any point in the fluid using Lamb's general solution for Stokes flow. While the original work addresses slightly deformed spheres to a linear order in deformation amplitude, we demonstrate that the first mapping, in combination with axisymmetric spectral modes (J. Fluid Mech., vol. 936, 2022, R1), can accommodate significant deformations to arbitrary orders of perturbation, and thus is not limited to slightly deformed spheres. Also, while first-order analysis is suitable for nearly spherical particles, second-order terms can provide a reasonable range for significantly higher deformations.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

Validating the theoretical work on Rayleigh–Taylor instability (RTI) through experiments with an exceptionally clean and well-characterized initial condition has been a long-standing challenge. Experiments were conducted to study the three-dimensional RTI of an SF$_6$–air interface at moderate Atwood numbers. A novel soap film technique was developed to create a discontinuous gaseous interface with controllable initial conditions. Spectrum analysis revealed that the initial perturbation of the soap film interface is half the size of an entire single-mode perturbation. The correlation between the initial interface perturbation and Atwood numbers was determined. Due to the steep and highly curved feature of the initial soap film interface, the early-time evolution of RTI exhibits significant nonlinearity. In the quasi-steady regime, various potential flow models accurately predict the late-time bubble velocities by considering the channel width as the perturbation wavelength. Differently, the late-time spike velocities are described by these potential flow models using the wavelength of the entire single-mode perturbation. These findings indicate that the bubble evolution is influenced primarily by the spatial constraint imposed by walls, while the spike evolution is influenced mainly by the initial curvature of the spike tip. Consequently, a recent potential flow model was adopted to describe the time-varying amplitude growth induced by RTI. Furthermore, the self-similar growth factors for bubbles and spikes were determined from experiments and compared with existing studies, revealing that a large amplitude in the initial soap film interface promotes the spike development.

The dynamics of a single highly elastic fibre settling under gravity in a very viscous fluid is studied numerically. We employ the bead model and multipole expansion of the Stokes equations, corrected for lubrication that is implemented in the precise Hydromultipole numerical codes. Four attracting regular dynamical modes of highly elastic fibres are found: two stationary shapes (one translating and the other rotating and translating), and two periodic oscillations around such shapes. The phase diagram of these modes is presented. It illustrates that the existence of each mode depends not only on the elasto-gravitation number but also on the fibre aspect ratio. Characteristic time scales, fibre deformation patterns and motion in the different modes are determined.

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.

Modons, or dipolar vortices, are common and long-lived features of the upper ocean, consisting of a pair of counter-rotating monopolar vortices moving through self-advection. Such structures remain stable over long times and may be important for fluid transport over large distances. Here, we present a semi-analytical method for finding fully nonlinear modon solutions in a multi-layer quasi-geostrophic model with arbitrarily many layers. Our approach is to reduce the problem to a multi-parameter linear eigenvalue problem which can be solved using numerical techniques from linear algebra. The method is shown to replicate previous results for one- and two-layer models and is applied to a three-layer model to find a solution describing a mid-depth propagating, topographic vortex.

Resistive tearing instabilities are common in fluids that are highly electrically conductive and carry strong currents. We determine the effect of stable stratification on the tearing instability under the Boussinesq approximation. Our results generalise previous work that considered only specific parameter regimes, and we show that the length scale of the fastest-growing mode depends non-monotonically on the stratification strength. We confirm our analytical results by solving the linearised equations numerically, and we discuss whether the instability could operate in the solar tachocline.

The background potential energy (BPE) is the only reservoir that double diffusive instabilities can tap their energy from when developing from an unforced motionless state with no available potential energy (APE). Recently, Middleton and Taylor linked the extraction of BPE into APE to the sign of the diapycnal component of the buoyancy flux, but their criterion can predict only diffusive convection instability, not salt finger instability. Here, we show that the problem can be corrected if the sign of the APE dissipation rate is used instead, making it emerge as the most fundamental criterion for double diffusive instabilities. A theory for the APE dissipation rate for a two-component fluid relative to its single-component counterpart is developed as a function of three parameters: the diffusivity ratio, the density ratio, and a spiciness parameter. The theory correctly predicts the occurrence of both salt finger and diffusive convection instabilities in the laminar unforced regime, while more generally predicting that the APE dissipation rate for a two-component fluid can be enhanced, suppressed, or even have the opposite sign compared to that for a single-component fluid, with important implications for the study of ocean mixing. Because negative APE dissipation can also occur in stably stratified single-component and doubly stable two-component stratified fluids, we speculate that only the thermodynamic theory of exergy can explain its physics; however, this necessitates accepting that APE dissipation is a conversion between APE and the internal energy component of BPE, in contrast to prevailing assumptions.

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.

Steep wave breaking on a vertical cylinder (a typical foundation supporting offshore wind turbines) will induce slam loads. Many questions on the important violent wave loading and the associated secondary load cycle remain unanswered. We use laboratory experiments with unidirectional waves to investigate the fluid loading on vertical cylinders. We use a novel three-phase decomposition approach that allows us to separate different types of nonlinearity. Our findings reveal the existence of an additional quasi-impulsive loading component that is associated with the secondary load cycle and occurs in the backwards direction against that of the incoming waves. This quasi-impulsive force occurs at the end of the secondary load cycle and close to the passage of the downward zero-crossing point of the undisturbed wave. Wavelet analysis showed that the impulsive force exhibits superficially similar behaviour to a typical wave-slamming event but in the reverse direction. To monitor the scattered wave field and extract run-up on the cylinder, we installed a four-camera synchronised video system and found a strong temporal correlation between the arrival time of the Type-II scattered wave onto the cylinder and the occurrence of this quasi-impulsive force. The temporal characteristics of this quasi-impulsive force can be approximated by the Goda wave impact model, taking the collision of the Type-II scattered waves at the rear stagnation point as the impact source.

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.