Flag flutter frequently features a marked difference between the onset speed of flutter and the speed below which flutter stops. The hysteresis tends to be especially large in experiments as opposed to simulations. This phenomenon has been ascribed to inherent imperfections of flatness in experimental samples, which are thought to inhibit the onset of flutter but have a lesser effect once a flag is already fluttering. In this work, we present an experimental confirmation for this explanation through motion tracking. We also visualize the wake to assess the potential contribution of discrete vortex shedding to hysteresis. We then mould our understanding of the mechanism of bistability and additional observations on flag flutter into a novel, observation-based, semiempirical model for flag flutter in the form of a single ordinary differential equation. Despite its simplicity, the model successfully reproduces key features of the physical system such as bistability, sudden transitions between non-fluttering and fluttering states, amplitude growth and frequency growth.

The effects of high-intensity, large-scale free stream turbulence on the aerodynamic loading and boundary layer flow field development on a NACA 0018 aerofoil model were studied experimentally using direct force measurements and particle image velocimetry at a chord Reynolds number of $7\times 10^4$. An active turbulence grid was used to generate free stream turbulence intensities of up to $16\,\%$ at integral length scales of the order of the aerofoil chord length. Relative to the clean flow condition with a free stream turbulence intensity of $0.1\,\%$, elevated levels of free stream turbulence intensity decrease the lift slope at low angles of attack, and increase the stall angle and maximum lift coefficient. At moderate angles of attack, high-intensity free stream turbulence causes large variations in the location of transition, with laminar flow occasionally persisting over $90\,\%$ of the chord length. At pre-stall angles of attack, high-intensity free stream turbulence causes intermittent massive separation. Variations in the extent of turbulence in the suction surface boundary layer are linked to fluctuations in effective angle of attack, suggesting that the observed variability in transition location is related to large-scale incoming flow disturbances impinging on the aerofoil model. A comparative analysis of the present results and those in previous studies for predominantly smaller integral length scales shows the importance of both the intensity and length scale of free stream turbulence on the flow development over the aerofoil.

The effects of Reynolds number across ${\textit{Re}}=1000$, $2500$, $5000$ and $10\,000$ on separated flow over a two-dimensional NACA0012 airfoil at an angle of attack of $\alpha =14^\circ$ are investigated through biglobal resolvent analysis. We identify modal structures and energy amplifications over a range of frequencies, spanwise wavenumbers, and values of the discount parameter, providing insights across various time scales. Using temporal discounting, we find that the shear-layer dynamics dominates over short time horizons, while the wake dynamics becomes the primary amplification mechanism over long time horizons. Spanwise effects also appear over long time horizons, sustained by low frequencies. The low-frequency and high-wavenumber structures are found to be dominated by elliptic mechanisms within the recirculation region. At a fixed angle of attack and across the Reynolds numbers, the response modes shift from wake-dominated structures at low frequencies to shear-layer-dominated structures at higher frequencies. The frequency at which the dominant mechanism changes is independent of the Reynolds number. Comparisons at a different angle of attack ($\alpha =9^\circ$) show that the transition from wake to shear-layer dynamics with increasing frequency only occurs if the unsteady flow is three-dimensional. We also study the dominant frequencies associated with wake and shear-layer dynamics across the angles of attack and Reynolds numbers, and confirm characteristic scaling laws from the literature.

We consider numerically a Lagrangian view of turbulent mixing in two-layer stably stratified parallel shear flow. By varying the ratio of shear layer depth to density interface thickness, these flows are prone to either a primary Kelvin–Helmholtz instability (KHI) or to a primary Holmboe wave instability (HWI). These instabilities are conventionally thought to mix qualitatively differently; by vortical ‘overturning’ of the density interface induced by KHI, or by turbulent ‘scouring’ on the edges of the density interface induced by HWI. By tracking Lagrangian particles in direct numerical simulations, so that the fluid buoyancy sampled along particle paths provides a particular Lagrangian measure of mixing, we investigate the validity of this overturning/scouring classification. The timing of mixing events experienced by particles inside and outside the interface is qualitatively different in simulations exhibiting KHI and HWI. The root mean square (r.m.s.) buoyancy for particles that start with the same buoyancy is actually larger for HWI-associated flows than for KHI-associated flows for the same bulk Richardson number $Ri_b$, implying heterogeneous mixing along particle paths for HWI. The number of particles starting close to the mid-plane of the interface which experience a change in sign in the local fluid buoyancy (and hence end up on the opposite side of the mid-plane after mixing) is compared for KHI and HWI in flows with various $Ri_b$. Perhaps surprisingly, for HWI with a large $Ri_b$, more than half of the particles that start near the mid-plane end up on the opposite side of the mid-plane.

The reconfiguration of flexible aquatic vegetation and the associated forces have been extensively studied under two-dimensional flow conditions – such as unidirectional currents, pure waves and co-directional wave–current flows. However, behaviour under more complex, orthogonal wave–current flows remains largely unexplored. In coastal environments, such orthogonal flows arise when waves propagate perpendicular to a longshore current. To improve understanding of how aquatic vegetation helps protect coastlines and attenuates waves, we extended existing effective-length scaling laws that were validated in pure currents, pure waves, and co-directional waves and currents to orthogonal wave–current conditions by introducing new definitions of the Cauchy number. Experiments were conducted in a wave–current basin, where cylindrical rubber stems were mounted on force transducers to measure hydrodynamic forces. Stem velocities were extracted from video recordings to compute the relative velocity between the flow and the stems. Incorporating the phase shift between flow and stem velocities into the force models significantly improved predictions. Comparison of predicted and measured forces showed good agreement for both pure wave and wave–current scenarios, underscoring the importance of phase shifts and velocity reduction for force estimation. Our hypothesised effective-length scaling parameters under wave–current conditions were validated, but with a higher scaling coefficient due to inertial effects from the larger material aspect ratio. These findings offer new insights into the hydrodynamics of flexible structures under complex coastal flow conditions.

We present a framework for parametric proper orthogonal decomposition (POD)-Galerkin reduced-order modelling (ROM) of fluid flows that accommodates variations in flow parameters and control inputs. As an initial step, to explore how the locally optimal POD modes vary with parameter changes, we demonstrate a sensitivity analysis of POD modes and their spanned subspace, respectively rooted in Stiefel and Grassmann manifolds. The sensitivity analysis, by defining distance between POD modes for different parameters, is applied to the flow around a rotating cylinder with varying Reynolds numbers and rotation rates. The sensitivity of the subspace spanned by POD modes to parameter changes is represented by a tangent vector on the Grassmann manifold. For the cylinder case, the inverse of the subspace sensitivity on the Grassmann manifold is proportional to the Roshko number, highlighting the connection between geometric properties and flow physics. Furthermore, the Reynolds number at which the subspace sensitivity approaches infinity corresponds to the lower bound at which the characteristic frequency of the Kármán vortex street exists (Noack & Eckelmann, J. Fluid Mech., 1994, vol. 270, pp. 297–330). From the Stiefel manifold perspective, sensitivity modes are derived to represent the flow field sensitivity, comprising the sensitivities of the POD modes and expansion coefficients. The temporal evolution of the flow field sensitivity is represented by superposing the sensitivity modes. Lastly, we devise a parametric POD-Galerkin ROM based on subspace interpolation on the Grassmann manifold. The reconstruction error of the ROM is intimately linked to the subspace-estimation error, which is in turn closely related to subspace sensitivity.

We analyse the long-time dynamics of trajectories within the stability boundary between laminar and turbulent square duct flow. If not constrained to a symmetric subspace, the edge trajectories exhibit a chaotic dynamics characterised by a sequence of alternating quiescent phases and intense bursting episodes. The dynamics reflects the different stages of the well-known near-wall streak–vortex interaction. Most of the time, the edge states feature a single streak with a number of flanking vortices attached to one of the four surrounding walls. The initially straight streak undergoes a linear instability and eventually breaks in an intense bursting event. At the same time, the downstream vortices give rise to a new low-speed streak at one of the neighbouring walls, thereby causing the turbulent activity to ‘switch’ from one wall to the other. If the edge dynamics is restricted to a single or twofold mirror-symmetric subspace, the bursting and wall-switching episodes become self-recurrent in time, representing the first periodic orbits found in square duct flow. In contrast to the chaotic edge states in the non-symmetric case, the imposed symmetries enforce analogue bursting cycles to simultaneously appear at two parallel opposing walls in a mirror-symmetric configuration. Both the localisation of turbulent activity to one or two walls and the wall-switching dynamics are shown to be common phenomena in marginally turbulent duct flows. We argue that such episodes represent transient visits of marginally turbulent trajectories to some of the edge states detected here.

The breakup and coalescence of particle aggregates confined at the interface of turbulent liquid layers are investigated experimentally and theoretically. In particular, we consider conductive fluid layers driven by Lorentz forces and laden with millimetre-scale floating particles. These form aggregates held together by capillary attraction and disrupted by the turbulent motion. The process is fully characterised by imaging at high spatio-temporal resolution. The breakup frequency $\varOmega$ is proportional to the mean strain rate and follows a power-law scaling $\varOmega \sim D^{3\text{/}2}$, where $D$ is the size of the aggregate, attributed to the juxtaposition of particle-scale strain cells. The daughter aggregate size distribution exhibits a robust U-shape, which implies erosion of small fragments as opposed to even splitting. The coalescence kernel $\varGamma$ between pairs of aggregates of size $D_{1}$ and $D_{2}$ scales as $\varGamma \sim ( D_{1} + D_{2} )^{2}$, which is consistent with gas-kinetic dynamics. These relations, which apply to regimes dominated both by capillary-driven aggregation and by drag-driven breakup, are implemented into the population balance equation for the evolution of the aggregate number density. Comparison with the experiments shows that the framework captures the observed distribution for aggregates smaller than the forcing length scale.

Experimental studies of natural convection in yield stress fluids have revealed transient behaviours that contradict predictions from viscoplastic models. For example, at a sufficiently large yield stress, these models predict complete motionlessness; below a critical value, yielding and motion onset can be delayed in viscoplastic models. In both cases, however, experiments observe immediate motion onset. We present numerical simulations of the transient natural convection of elastoviscoplastic (EVP) fluids in a square cavity with differentially heated side walls, exploring the role of elasticity in reconciling theoretical predictions with experimental observations. We consider motion onset in EVP fluids under two initial temperature distributions: (i) a linear distribution characteristic of steady pure conduction, and (ii) a uniform distribution representative of experimental conditions. The Saramito EVP model exhibits an asymptotic behaviour similar to the Kelvin-Voigt model as $t\to 0^+$, where material behaviour is primarily governed by elasticity and solvent viscosity. The distinction between motion onset and yielding, a hallmark of EVP models, is the key feature that bridges theoretical predictions with experimental observations. While motion onset is consistently immediate (as seen in experiments), yielding occurs with a delay (as predicted by viscoplastic models). Scaling analysis suggests that this delay varies logarithmically with the yield stress and is inversely proportional to the elastic modulus. The intensity of the initial pre-yield motion increases with higher yield stress and lower elastic modulus. The observed dynamics resemble those of under- and partially over-damped systems, with a power-law fit providing an excellent match for the variation of oscillation frequency with the elastic modulus.

Nearly fifty years ago, Roberts (1978) postulated that the Earth’s magnetic field, which is generated by turbulent motions of liquid metal in its outer core, likely results from a subcritical dynamo instability characterised by a dominant balance between Coriolis, pressure and Lorentz forces (requiring a finite-amplitude magnetic field). Here, we numerically explore subcritical convective dynamo action in a spherical shell, using techniques from optimal control and dynamical systems theory to uncover the nonlinear dynamics of magnetic field generation. Through nonlinear optimisation, via direct-adjoint looping, we identify the minimal seed – the smallest magnetic field that attracts to a nonlinear dynamo solution. Additionally, using the Newton-hookstep algorithm, we converge stable and unstable travelling wave solutions to the governing equations. By combining these two techniques, complex nonlinear pathways between attracting states are revealed, providing insight into a potential subcritical origin of the geodynamo. This paper showcases these methods on the widely studied benchmark of Christensen et al. (2001, Phys. Earth Planet. Inter., vol. 128, pp. 25–34), laying the foundations for future studies in more extreme and realistic parameter regimes. We show that the minimal seed reaches a nonlinear dynamo solution by first approaching an unstable travelling wave solution, which acts as an edge state separating a hydrodynamic solution from a magnetohydrodynamic one. Furthermore, by carefully examining the choice of cost functional, we establish a robust optimisation procedure that can systematically locate dynamo solutions on short time horizons with no prior knowledge of its structure.

Induced diffusion (ID), an important mechanism of spectral energy transfer due to interacting internal gravity waves (IGWs), plays a significant role in driving turbulent dissipation in the ocean interior. In this study, we revisit the ID mechanism to elucidate its directionality and role in ocean mixing under varying IGW spectral forms, with particular attention to deviations from the standard Garrett–Munk spectrum. The original interpretation of ID as an action diffusion process, as proposed by McComas et al., suggests that ID is inherently bidirectional, with its direction governed by the vertical-wavenumber spectral slope $\sigma$ of the IGW action spectrum, $n \propto m^\sigma$. However, through the direct evaluation of the wave kinetic equation, we reveal a more complete depiction of ID, comprising both a diffusive and a scale-separated transfer rooted in the energy conservation within wave triads. Although the action diffusion may reverse direction depending on the sign of $\sigma$ (i.e. red or blue spectra), the net transfer by ID consistently leads to a forward energy cascade at the dissipation scale, contributing positively to turbulent dissipation. This supports the viewpoint of ID as a dissipative mechanism in physical oceanography. This study presents a physically grounded overview of ID, and offers insights into the specific types of wave–wave interactions responsible for turbulent dissipation.

Using direct numerical simulations, we systematically investigate the inner-layer turbulence of a turbulent vertical buoyancy layer (a model for a vertical natural convection boundary layer) at a constant Prandtl number of $0.71$. Near-wall streaky structures of streamwise velocity fluctuations, synonymous with the buffer layer streaks of canonical wall turbulence, are not evident at low and moderate Reynolds numbers (${\textit{Re}}$) but manifest at high ${\textit{Re}}$. At low ${\textit{Re}}$, the turbulent production in the near-wall region is negligible; however, this increases with increasing ${\textit{Re}}$. By using domains truncated in the streamwise, spanwise and wall-normal directions, we demonstrate that the turbulence production in the near-wall region at moderate and high ${\textit{Re}}$ is largely independent of large-scale motions and outer-layer turbulence. On a fundamental level, the near-wall turbulence production is autonomous and self-sustaining, and a well-developed bulk is not needed to drive the inner-layer turbulence. Near-wall streaks are also not essential for this autonomous process. The type of thermal boundary condition only marginally influences the velocity fluctuations, revealing that the turbulence dynamics are primarily governed by the mean-shear induced by the buoyancy field and not by the thermal fluctuations, despite the current flow being solely driven by buoyancy. In the inner layer, the spanwise wavelength of the eddies responsible for positive shear production is remarkably similar to that of canonical wall turbulence at moderate and high ${\textit{Re}}$ (irrespective of near-wall streaks). Based on these findings, we propose a mechanistic model that unifies the near-wall shear production of vertical buoyancy layers and canonical wall turbulence.

We derive a mathematical model for the overflow fusion glass manufacturing process. In the limit of zero wedge angle, the model leads to a canonical fluid mechanics problem in which, under the effects of gravity and surface tension, a free-surface viscous flow transitions from lubrication flow to extensional flow. We explore the leading-order behaviour of this problem in the limit of small capillary number, and find that there are four distinct regions where different physical effects control the flow. We obtain leading-order governing equations, and determine the solution in each region using asymptotic matching. The downstream behaviour reveals appropriate far-field conditions to impose on the full problem, resulting in a simple governing equation for the film thickness that holds at leading order across the entire domain.

We explore the mechanisms and regimes of mixing in yield-stress fluids by simulating the stirring of an infinite, two-dimensional domain filled with a Bingham fluid. A cylindrical stirrer moves along a circular path at constant speed, with the path radius fixed at twice the stirrer diameter; the domain is initially quiescent and marked by a passive dye in the lower half. We first examine the mixing process in Newtonian fluids, identifying three key mechanisms: interface stretching and folding around the stirrer’s path, diffusion across streamlines and dye advection and interface stretching due to vortex shedding. Introducing yield stress leads to notable mixing localisation, manifesting through three mechanisms: advection of vortices within a finite distance of the stirrer, vortex entrapment near the stirrer and complete suppression of vortex shedding at high yield stresses. Based on these mechanisms, we classify three distinct mixing regimes: (i) regime SE, where shed vortices escape the central region, (ii) regime ST, where shed vortices remain trapped near the stirrer and (iii) regime NS, where no vortex shedding occurs. These regimes are quantitatively distinguished through spectral analysis of energy oscillations, revealing transitions and the critical Bingham and Reynolds numbers. The transitions are captured through effective Reynolds numbers, supporting the hypothesis that mixing regime transitions in yield-stress fluids share fundamental characteristics with bluff-body flow dynamics. The findings provide a mechanistic framework for understanding and predicting mixing behaviours in yield-stress fluids, suggesting that the localisation mechanisms and mixing regimes observed here are archetypal for stirred-tank applications.

Transonic buffet presents time-dependent aerodynamic characteristics associated with shock, turbulent boundary layer and their interactions. Despite strong nonlinearities and a large degree of freedom, there exists a dominant dynamic pattern of a buffet cycle, suggesting the low dimensionality of transonic buffet phenomena. This study seeks a low-dimensional representation of transonic airfoil buffet at a high Reynolds number with machine learning. Wall-modelled large-eddy simulations of flow over the OAT15A supercritical airfoil at two Mach numbers, $M_\infty = 0.715$ and 0.730, respectively producing non-buffet and buffet conditions, at a chord-based Reynolds number of ${Re} = 3\times 10^6$ are performed to generate the present datasets. We find that the low-dimensional nature of transonic airfoil buffet can be extracted as a sole three-dimensional latent representation through lift-augmented autoencoder compression. The current low-order representation not only describes the shock movement but also captures the moment when the separation occurs near the trailing edge in a low-order manner. We further show that it is possible to perform sensor-based reconstruction through the present low-dimensional expression while identifying the sensitivity with respect to aerodynamic responses. The present model trained at ${Re} = 3\times 10^6$ is lastly evaluated at the level of a real aircraft operation of ${Re} = 3\times 10^7$, exhibiting that the phase dynamics of lift is reasonably estimated from sparse sensors. The current study may provide a foundation towards data-driven real-time analysis of transonic buffet conditions under aircraft operation.

Can a fish-like body swim in a perfect fluid – one that is purely inviscid and does not release vorticity? This question was raised by Saffman over fifty years ago, and he provided a positive answer by demonstrating a possible solution for an inhomogeneous body. In this paper, we seek to determine a suitable deformation for oscillatory fish swimming that enables slight locomotion in a perfect fluid, relying solely on tail flapping motion. This swimming style, typical of carangiform and thunniform species, allows for a separate analysis of the tail’s interaction with the surrounding fluid. As a preliminary approach, the tail is approximated as a rigid plate with prescribed heave and pitch motions, while the presence of a virtual body placed in front is considered to evaluate the locomotion. Analytical solutions provide exact results while avoiding singular behaviour at sharp edges. A phase shift is shown to be strictly necessary for generating locomotion. A more refined approximation of a real fish is achieved by modelling the tail as a flexible foil, connected to the main body via a torsional spring with tuneable stiffness at the peduncle. While the heave motion remains prescribed, the pitch amplitude and phase are passively determined by flow interaction. A plausible solution reveals an optimal stride length as a function of dimensionless stiffness, driven by resonance phenomena. A small structural damping must be considered to induce a phase shift – essential for self-propulsion in the absence of vorticity release.

The spatio-temporal evolution of very large-scale coherent structures, also known as superstructures, is investigated in both smooth- and rough-wall boundary layers by means of direct numerical simulations up to a frictional Reynolds number of ${\textit{Re}}_\tau = 3\,150$. One smooth-wall and four rough-wall cases are considered, all developing over a region as long as $\sim$60 times the incoming boundary-layer thickness in the streamwise direction. Bio-inspired, biofouling-type topographies are employed for the rough-wall cases, following the previous work of Womack et al. (2022 J. Fluid Mech. vol. 933, p. A38) and Kaminaris et al. (2023 J. Fluid Mech. vol. 961, p. A23). We utilise three-dimensional time series, as well as multiple two-point correlation functions along the boundary layer to capture the detailed length- and time-scale evolution of the superstructures. The results suggest that the presence of roughness significantly amplifies both the strength and the streamwise growth rate of superstructures. Interestingly, however, their ratios relative to the local boundary-layer thickness, $\mathscr{L}_{\!x}/\delta$ and $\mathscr{L}_z/\delta$, remain constant and independent of the streamwise coordinate, indicating that such scaled length scales might constitute a possible flow invariant. Volumetric correlations revealed that all cases induce structures inclined with respect to the mean-flow direction, with those over the rough-wall topographies exhibiting steeper inclination angles. Finally, via proper orthogonal decomposition, pairs of counter-rotating roll modes were detected and found to flank the high- and low-speed superstructures, supporting the conjecture in the literature regarding the mechanisms responsible for the lateral momentum redistribution. The latter also suggests that the way momentum organises itself in high Reynolds number wall-bounded flows might be independent of the roughness terrain underneath.

This study quantitatively investigates the two-dimensional pseudosteady shock refraction at an inclined air–water interface, referred to as the water wedge, in the weak and strong incident shock strength groups. Numerical simulations are employed to validate the predicted refraction sequences from a previous study (Anbu Serene Raj et al. 2024 J. Fluid Mech. 998, A49). A distinctive irregular refraction pattern, referred to as the bound precursor refraction with a Mach reflection, is numerically validated in the weak shock group. Based on the numerical simulations, an enhanced formulation is proposed to determine the sonic line of the incident flow Mach number ($M_b$) in water, thereby providing an appropriate transition condition for an irregular refraction with a Mach reflection to a free precursor refraction with a Mach reflection transition. Furthermore, comparative studies on solid and water wedges of wedge angle $20^\circ$ reveal discernible differences in the shock reflection patterns. The interplay of the energy dissipation due to the transmitted shock wave and the Richtmyer–Meshkov instability at the air–water interface results in the variation of the triple-point trajectory and transition angles between single Mach reflection (SMR) to transitional Mach reflection (TMR) occurring in air.

We develop a weakly nonlinear theory to revisit the water hammer phenomenon resulting from slow valve manoeuvres. The hydraulic head at the valve is known to be nonlinearly coupled with the flow velocity via a relation derived from Bernoulli’s principle, so that solutions have been so far obtained only via numerical models. The governing equations and boundary conditions indeed yield a nonlinear boundary-value problem, which is here solved using a perturbation approach, Laplace transform and complex analysis. We obtain space- and time-dependent analytical solutions in all of the pipe and validate our results by comparison with standard numerical methods (i.e. Allievi’s method) for determining the exact behaviour of the hydraulic head at the valve. Additionally, we derive algebraic practically relevant closed form expressions for predicting the maximum and minimum hydraulic head values during both valve closure and opening manoeuvres.

Solid atmospheric particles, such as ice crystals, pollen, dust, ash and microplastics, strongly influence Earth’s climate, ecosystems and air quality. Previous studies have typically relied on analytical models valid only for very small particles or experiments in liquids, where the particle-to-fluid density ratio $R$ is much lower than values encountered in the atmosphere. We combine a novel experimental set-up with particle-resolved direct numerical simulations to study the settling of sub-millimetre ellipsoids in still air. Particle shapes span elongation and flatness values $ 0.2 \leqslant {\textit{EL}}, {\textit{FL}} \leqslant 1.0$ at a density ratio $ R = 1000$ and particle Reynolds numbers $ 2.1 \lt {\textit{Re}}_{\!p} \lt 4.5$, a regime well below the onset of wake-induced instabilities. Nonetheless, we observe unexpectedly rich dynamics: all non-spherical particles exhibit damped oscillatory motion, and some triaxial ellipsoids follow fully three-dimensional, non-planar trajectories due to rotation about all three axes. Simulations at lower density ratios ($ R = 10, 100$) confirm that these behaviours are driven by strong lateral forces happening only at $R=1000$. Key settling characteristics exhibit nonlinear and non-trivial dependencies on shape. In the two-dimensional phase space of elongation and flatness, settling velocity is symmetric about the principal diagonal ($ {\textit{EL}} = {\textit{FL}}$), while oscillation frequency and damping rate show symmetry about the anti-diagonal. Flatness strongly influences pressure drag, while elongation governs lateral drift and swept volume, which can reach up to ten times the particle diameter and four times the volume-equivalent sphere, respectively.