This paper investigates the mean flow asymmetry about the meridional plane in crossflow over a 6 : 1 prolate spheroid using high-fidelity numerical simulations. A series of direct numerical simulations are performed at diameter-based Reynolds number $\textit{Re}_{\!D} = 3.0 \times 10^3$ over a range of angles of attack. We identify a critical angle of attack for the onset of mean flow asymmetry between $40^\circ$ and $42^\circ$. In cases where asymmetry eventually develops, the flow initially remains symmetric for an extended period before turbulent fluctuations in the wake perturb the symmetry. As wake turbulence becomes more vigorous at higher Reynolds numbers, this observation suggests a reduced critical angle of attack – an expectation confirmed by simulations at $\textit{Re}_{\!D} = 6.0 \times 10^3$. To investigate the mechanism responsible for the asymmetry, we propose a new measure of mean asymmetry and derive a corresponding transport equation from the Navier–Stokes equations. This formulation identifies the production and destruction terms governing the evolution of asymmetry. Our analysis of the equation reveals that the generation mechanism is primarily inviscid, suggesting that the findings at low Reynolds number may extend to higher Reynolds numbers. Finally, we present spectral analyses of the force and moment histories at $45^\circ$ angle of attack, revealing two dominant frequencies and their physical origin, and quantifying inter-scale interactions by applying amplitude modulation analysis to the force and moment signals.

Experimental investigation is presented to elucidate flow structures and corresponding frequencies on finite span, cantilevered wings as functions of sweep angle and taper ratio. A detailed parameters sweep of planform geometry varied the leading and trailing edge sweep angles of NACA 0015 wings with a semi-aspect ratio of 2. The experiments were performed at Reynolds numbers O($10^5$) and angles of attack $12^\circ$–$22^\circ$. The experiments included flow visualisations, volumetric flow measurements, time-resolved measurements at selected spanwise planes and aerodynamic loads. This is the first time, to the best knowledge of the authors, that multiple configurations were tested under the same exact conditions. The correlation between three-dimensional (3-D) Reynolds stress distributions and 3-D flow separation is presented. Moreover, spectral content reveals modes that vary along the span, as well as for different planforms. For all wings, at the locations of largest reversed flow, power spectral density (PSD) peaks were seen at $0.1\lt St\lt 0.2$, corresponding to vortex shedding. At spanwise locations near surface spirals the PSD exhibits peaks at lower frequencies of St = O(0.01) due to focus point wandering. The mean flow fields presented here show similarities to previous numerical simulation findings on similar geometries at Reynolds number O($10^2$). Moreover, the spanwise location of the largest magnitude of turbulent kinetic energy corresponds to the location of the most amplified mode at a Reynolds number of 400 found previously. The present research, complemented by previous low-Reynolds-number work, provides fundamental insights into global flow structures on multiple finite span wings, their corresponding spectral content and the effect on their aerodynamic performance.

We numerically study the flow past an azimuthally oscillating cylinder at Reynolds number ${\textit{Re}}=250$ to analyse the three-dimensionalities observed in the recent experiments of Bhattacharyya et al. (J. Fluid Mech., vol. 950, 2022, p. A10). Specifically, we focus on the two newly discovered three-dimensional instability modes, referred to as modes Z and Y in the experiments, by suitably varying the cylinder oscillation amplitude and forcing frequency. Our direct numerical simulations (DNS) visually confirm the unique honeycomb-like structure of mode Y, also matching its spanwise wavelength, while mode Z, also referred to as mode D elsewhere, is not found at the expected parametric space but at an oscillation amplitude three times higher. Spectral proper orthogonal decomposition modes extracted from the DNS data reveal the near-wake dynamics of mode Y to be modulated by the forcing frequency and its subharmonics. Mode D/Z is found to be strongly correlated with a two-dimensional modal regime, especially at the forcing frequency, its subharmonic and higher harmonics. Mode Y does not show significant correlations with this two-dimensional flow except at a low frequency and only at its far wake. The honeycomb nature of mode Y is a result of its relatively higher forcing frequency causing multiple vortex sheddings over a rather compact space. Higher cylinder oscillation amplitude increases the overall drag, with the two-dimensional flow regimes generally yielding lower drag and lift. The results here suggest mode Y to be a unique three-dimensional mode of azimuthally oscillating cylinders and mode D/Z to be merely an intermediate state with the cylinder wake transitioning from the classical three-dimensional mode B to a two-dimensional state.

In this paper, the evaporation of neighbouring multi-component droplets or rivulets – often found in applications such as inkjet printing, spray cooling and pesticide delivery – is studied numerically and theoretically. The proximity induces a shielding effect that reduces individual evaporation rates and disrupts the symmetry of both the concentration profile and the flow field in the liquids. We examine how the symmetry of flow and concentration fields is affected by key parameters, namely the contact angle, the inter-droplet (or inter-rivulet) distance and the magnitude of surface tension gradient forces (i.e. the Marangoni number). We focus on binary mixtures, such as water and 1,2-hexanediol, where only one component evaporates and evaporation is slow, thereby allowing simplifications to the governing equations. To manage the complexity of the full three-dimensional droplet problem, we begin with a two-dimensional model of neighbouring rivulets. Solving the complete transient equations for rivulets with pinned contact lines and fixed inter-rivulet distance reveals that the asymmetry – quantified by the position of the interfacial stagnation point of the flow – diminishes over time. Using a validated quasi-stationary model, we find, with increasing contact angle and inter-rivulet distance, that the stagnation point migrates closer to the centre, yet it remains unaffected by the Marangoni number. A simplified lubrication model applied to droplets shows similar dependencies on contact angle and distance, although here the stagnation point appears to vary with the Marangoni number. We attribute this dependence to the additional azimuthal flow in droplets, leading to a nonlinear evolution of the concentration and therefore a non-trivial dependence of the symmetry on the Marangoni number.

We use the principles of non-equilibrium thermodynamics to rigorously formulate the transport equations for granular systems consisting of oriented particles. The state variables are taken to be the density, velocity, thermal temperature, granular temperature (particles agitation) and the orientation tensor. The evolution of the state variables is governed by the associated balance laws in terms of fluxes. The contributions of the granular agitation energy and orientation to entropy are introduced into the Gibbs equation. The balance of entropy is used to identify the entropy production as the product of thermodynamics forces and fluxes. Using classical linear non-equilibrium thermodynamics the fluxes are considered to be linear in the thermodynamic forces. The Onsager–Casimir reciprocal relations and the representation theorem of isotropic tensors provide further restrictions that simplify the formulation. The non-negative entropy production requirement is satisfied by restricting the matrix of phenomenological coefficients to be positive semidefinite. Similarly the boundary conditions are constructed. The transport coefficients are then determined by comparison with available results from the granular kinetic theory of spherical particles and other available results for oriented particles. It is shown that not only these results are well captured, but also the formulation provides a framework for further generalization. The significant contribution of this work is the rigorous formulation of a physically admissible generalization to granular gases of oriented particles which reveals the role of the orientation in the transport equations and identifies couplings that might otherwise be omitted.

An important category of microscale fluid–structure interactions concerns how flexible fibres deform and interact with flows. Many experimental and numerical studies have focused on the shape dynamics of fibres in linear shear flows. Here, instead, we consider a fully three-dimensional background flow with non-constant vorticity and study the shape evolution of fibres in a zero-Reynolds-number analogue of a Burgers vortex. This flow is created by the superposition of regularised singularities of the Stokes equations. Using a Kirchhoff rod model with regularised Stokeslet segments that track both curvature and torsion evolution along the fibre, we observe novel three-dimensional deformations. The shape dynamics depends on two non-dimensional parameters: an elastoviscous number and the ratio of vortex core diameter to fibre length. We focus on the special case of fibre excursions when the fibre is placed in the horizontal plane of symmetry, centred at the vortex core. We reveal robust orbits where fibres spin about the z axis as they deform, but ultimately straighten out and reach a vertical equilibrium state. Our model demonstrates that the fibre flexibility influences the time it takes to complete this orbit, with flexible fibres reaching equilibrium sooner than their stiffer counterparts. In addition, we demonstrate that fibres placed asymmetrically within this fully three-dimensional background flow exhibit a wide array of shape evolutions, including helical buckling.

Richtmyer–Meshkov instability (RMI) at a single-mode interface separating an inert gas (N$_2$) and a reactive gas mixture (H$_2$/O$_2$/Xe) under reshock conditions is numerically investigated using a newly developed compressible reactive Navier–Stokes solver. The solver employs the Kéromnès mechanism (10 species, 21 reactions) for combustion modelling and a dual-flux algorithm to suppress numerical oscillations at material interfaces, demonstrating high accuracy across a wide range of benchmark tests. By systematically varying incident shock Mach numbers, we identify four distinct evolution regimes: an inert regime (${\textit{Ma}} \lt 1.80$), characterised by negligible combustion effects on interface evolution; a deflagration regime ($1.80 \lt Ma \lt 1.86$), marked by strong coupling between interface dynamics and combustion through sustained interactions; a detonation regime ($1.86 \lt Ma \lt 2.50$), where rapid transition to detonation leads to moderate coupling; and an immediate detonation regime (${\textit{Ma}} \gt 2.50$), where detonation occurs directly after incident shock impact, modifying interface evolution from the outset through intense heat release and pressure waves. Mixing width and mixing level are most significantly enhanced in the deflagration regime due to prolonged combustion-flow interactions, while cases with higher Mach numbers show reduced mixing due to rapid combustion completion. Heat release and enstrophy also display clear regime-dependent evolution behaviour: maximum heat release occurs in the detonation regime, while peak enstrophy is observed in the deflagration regime. A clear correlation is observed between the Damköhler number ($Da$), which represents the ratio of hydrodynamic to chemical time scales, and the flow regimes: for ${\textit{Ma}} \lt 1.80$, $Da \lt 1$ indicates negligible coupling; at ${\textit{Ma}} = 1.83$, $Da \approx 1$ reflects sustained coupling; and for ${\textit{Ma}} \gt 2.00$, $Da \gt 1$ denotes strong early coupling. This correlation provides a theoretical basis for interpreting the distinct regimes and guiding the modulation of reactive RMI.

This paper examines two-dimensional liquid curtains ejected from a narrow horizontal outlet at an angle to the vertical. Curtains are characterised by the Froude number ${\textit{Fr}}=U/ ( gH ) ^{1/2}$, Reynolds number ${\textit{Re}}=UH/\nu$ and Weber number ${\textit{We}}=\rho U^{2}H/\sigma$, where $U$ is the ejection velocity, $g$ the gravity, $H$ the outlet’s half-width, $\nu$ the kinematic viscosity and $\sigma$ the surface tension. It is assumed that ${\textit{Fr}}\gg 1$ (so that the radius of the curtain’s curvature due to gravity exceeds $H$), ${\textit{Re}}\ll 1$ (viscosity is strong) and ${\textit{We}}\sim 1$ (surface tension is on par with inertia). It is shown that steady oblique curtains exist only subject to a constraint of the form ${\textit{We}}\gt f({\textit{Fr}}^{2}{\textit{Re}})$, which is more restrictive than the previously known constraint ${\textit{We}}\gt 1$. Thus, sufficiently strong viscosity and/or surface tension eliminate the steady regime and make the curtain evolve – typically, rotate around the outlet, eventually producing the teapot effect.

Motivated by applications to underwater explosions and volcanic eruptions, this paper considers the evolution of an initial pressure disturbance in the ocean, including effects due to the dynamic and static compression of water and the free surface. In order to solve the equations of motion of a linear compressible ocean, a special inner product is introduced, which allows us to apply self-adjoint operator theory. What results is a Hilbert space in which the acoustic–gravity modes are orthogonal in the generalised sense. This allows the time-domain evolution of the free surface and subsurface pressure field resulting from an initial disturbance to be calculated. Our simulations show initial radial propagation of the pressure pulse and subsequent reflection from the water's surface and the rigid ocean floor, eventually leading to horizontal propagation away from the source point. The solutions with and without the inclusion of the static compression are compared, and the effect of static compression is shown to be small but not negligible.

Fluid mixture models are essential for describing a wide range of physical phenomena, including wave dynamics and spinodal decomposition. However, there is a lack of consensus in the modelling of compressible mixtures, with limited connections between different classes of models. On the one hand, existing compressible two-phase flow models accurately describe wave dynamics, but do not incorporate phase separation mechanisms. On the other hand, phase-field technology in fluid dynamics consists of models incorporating spinodal decomposition; however, a general phase-field theory for compressible mixtures remains largely undeveloped. In this paper we take an initial step toward bridging the gap between compressible two-phase flow models and phase-field models by developing a theory for compressible, isothermal N-phase mixtures. Our theory establishes a system of reduced complexity by formulating N mass balance laws alongside a single momentum balance law, thereby naturally extending the Navier–Stokes Korteweg model to N phases and providing the Navier–Stokes Cahn–Hilliard/Allen–Cahn model for compressible mixtures. Key aspects of the framework include its grounding in continuum mixture theory and its preservation of thermodynamic consistency despite its reduced complexity.

Rapid granular free-surface flows on inclined planes can develop secondary vortices aligned with the dominant flow direction. The reason for their formation remains a subject of research, but plausible mechanisms include instabilities driven by (i) dilatation/compressibility, (ii) normal stress differences and (iii) a self-induced Raleigh–Taylor instability caused by segregation of large–dense and small–light particles. In this paper, a set of novel experiments are performed with large and small particles (of the same bulk density), which form longitudinal stripes due to a combination of secondary recirculation and particle-size segregation. A conceptual model is formulated, in which large particles concentrate in the downwelling sections, small particles concentrate in the upwelling sections and a breaking-size-segregation wave separates the two pure phases from one another. In each secondary vortex, the breaking waves allow the large and small particles to continuously recirculate. Assuming that a series of counter-rotating vortices exist, it is shown that this internal cross-slope structure emerges naturally from solving the gravity-shear-driven segregation-advection equations. When viewed from above, this generates a series of alternating bands of large and small particles, that are sharply separated from one another and are aligned with the downslope direction. Each complete stripe (measured from centre to centre of each large band) is formed by two counter-rotating secondary vortices. Despite the apparent order of the steady-state stripes, it is shown that the individual large and small particle paths form complex interpenetrating co-rotating sub-vortices as they avalanche downslope.

This paper investigates the physical origins of pressure fluctuations on the stationary shroud wall of a mixed-flow pump. A novel ‘triple source model’ is developed and applied to experimental validated stress-blended eddy simulations. The model decomposes stationary-frame pressure fluctuations into three distinct rotating-frame components to disentangle complex tip leakage vortex (TLV) interactions: (i) kinematic ‘non-uniform fluctuation’ ($p_{\textit{NUF}}$) from the steady blade sweep, (ii) dynamic ‘flow synchronous fluctuation’ ($p_{\textit{FSF}}$) phase-locked to rotation, and (iii) ‘flow asynchronous fluctuation’ ($p_{\textit{FAF}}$) from all non-phase-locked phenomena. Analysis reveals that shroud unsteadiness is over 90 % dominated by the synchronous components along the TLV trajectory. Crucially, the model uncovers a counter-intuitive destructive interference mechanism between the kinematic sweep $p_{\textit{NUF}}$ and the dynamic response $p_{\textit{FSF}}$, with local cross-correlation coefficient –0.26, explaining how dynamic instabilities can dampen the steady pressure footprint. Source-term analysis of the pressure Poisson equation establishes a complete causal chain from specific velocity field interactions to pressure signatures: (i) the non-uniform fluctuation is kinematically driven by the mean momentum flux from blade loading, contributing 52.27 % to the local pressure asymmetry; (ii) the flow synchronous fluctuation is generated by periodic vortex–turbulence interaction, contributing 80.22 % of its total source; (iii) and the asynchronous broadband pressure is sourced from the canonical turbulent cascade, contributing 79.33 % of its total source. Spatial correlations confirm the TLV as the common physical nexus for all components. This work establishes a quantitative diagnostic framework that moves beyond qualitative vortex observation, providing a physical basis for the targeted mitigation of turbomachinery unsteadiness.

This paper employs the ensemble-based data assimilation method to develop a closed-form correction term for the Spalart–Allmaras (S–A) turbulence model to enhance predictive accuracy in separated flows through model-form uncertainty reduction. A compact radial-basis-function expression is proposed as correction model to supersede conventional modification procedures in classic field inversion and machine learning frameworks, achieving computational economy through spatially bounded correction regions. The correction model is derived via the Ensemble Kalman method with effective utilisation of synthesised observations based on the multi-fidelity data aggregation. The modified compact expression trained on a single case is systematically evaluated against unseen separation scenarios and the results show that the developed model can improve the prediction accuracy of flow separation in different validation cases, and the effectiveness of the method is verified. Compared with other black-box models, the correction based on the radial-basis-function form offers reduced complexity and high suitability for direct integration into numerical solvers. This approach facilitates cost-effective data assimilation and enables dynamic adaptation of the correction, thereby enhancing the generalisation capability for similar flow separation conditions.