In this study, we investigate the dynamics of a viscoplastic fluid placed above a Newtonian fluid in an inclined, long tube. The two fluids are miscible and have a density difference, with the viscoplastic fluid always denser. The experimental set-up allows varying inclination angles. Initially, the fluids are separated by a gate valve; upon opening it, distinct flow behaviours are observed, captured using cameras. The flow regimes depend on the interplay between buoyancy (due to the density difference), the yield stress of the viscoplastic fluid and the inclination angle. In tilted tubes, exchange flow with a slumping regime is observed, while in vertical configurations, a stable finger front forms at high values of the yield number ($Y$), representing the ratio of yield stress to buoyancy stress. As $Y$ decreases, helical, disconnected and slug finger regimes emerge. We investigate these flow regimes and quantify their transition boundaries, focusing on finger length and front velocity. Numerically, we develop a three-dimensional model based on the Herschel–Bulkley rheology, implementing the Papanastasiou regularisation and the volume-of-fluid method within the finite-volume framework of OpenFOAM. The simulations provide insights into velocity profiles, finger thickness and the distribution of yielded and unyielded regions. Numerical results indicate the finger tip becomes increasingly eccentric as the pipe is inclined, with a pronounced rise in eccentricity for inclinations exceeding $30^\circ$. These results are particularly relevant to industries like oil and gas, improving well integrity and protecting groundwater and the atmosphere by addressing cementing and sealing challenges.

We present a framework for large eddy simulations (LES) with immersed boundaries (IBs) to simulate high-Reynolds-number flows over complex walls. In this approach, which we call immersed boundary-modelled LES (IBMLES), we volume filter the Navier–Stokes equations to derive the IB bodyforce. This also gives rise to the subfilter-scale (SFS) stress and residual viscous stress tensors, the latter of which is closed, and we expect the SFS stress can be closed with existing models. We show that the IB bodyforce can be closed by modelling the wall-slip velocity and provide two such models. The first is an algebraic model based on volume filtering the Van Driest velocity profile, and the second is a slip-length model. We perform an a priori analysis by volume filtering direct numerical simulation (DNS) data of turbulent channel flow at friction Reynolds number $ \textit{Re}_{\tau }=5200$ to inform these models and investigate the behaviour of the other terms in this formulation. We find the streamwise wall-slip velocity is significant and both models show good agreement with volume-filtered DNS data on average. Slip velocity is non-uniform and retains a signature of inner or large-scale flow structures depending on filter size. The SFS terms are analogous to those in traditional LES and can likely be modelled with existing SFS models such the dynamic Smagorinsky model. Residual viscous stress is significant, so it must be considered in IBMLES. We perform filtering with multiple filter types and find there is little sensitivity to the choice of filter kernel so long as it abides by the assumptions given in this framework. Based on these results, we propose a dynamic procedure for determining the slip length that will be assessed in future studies.

We present a numerical study of three-dimensional gravity-capillary standing waves by using cubic and quintic truncated Hamiltonian formulations and the Craig–Sulem expansion of the Dirichlet–Neumann operator. The resulting models are treated as triply periodic boundary-value problems and solved via a spatio-temporal collocation method without executing initial-value calculations. This approach avoids the numerical stiffness associated with surface tension, and numerical instabilities arising from time integration. We reduce the number of unknowns significantly by exploiting the spatio-temporal symmetries for three types of symmetric standing waves. Comparisons with existing asymptotic and numerical results illustrate excellent agreement between the models and the full potential-flow formulation. We investigate typical bifurcations and standing waves that feature square, hexagonal and more complex flower-like patterns under the three-wave resonance. These solutions are generalisations of the classical Wilton ripples. Temporal simulations of the computed three-dimensional standing waves exhibit perfect periodicity and reveal an instability mechanism based on the previous reported oblique instability in two-dimensional standing waves.

Building upon the theoretical investigation of fully developed porous-channel flow by Nield et al. (J. Fluid Mech. vol. 322, 1996, 201–214), the present study examines hydrodynamically developing flow in anisotropic porous channels to better characterise realistic transport behaviour. A variational formulation for the steady incompressible Darcy–Brinkman–Forchheimer (DBF) equations is derived, where the Darcy, Brinkman and Forchheimer terms represent porous drag, viscous shear effects and nonlinear inertial resistance, respectively. The existence and uniqueness of solution are established using the Browder–Minty theorem together with the inf-sup condition to ensure the uniqueness of the pressure field. The porous medium is represented through a spatially varying anisotropic permeability tensor to account for directional resistance effects. A physics-informed neural network (PINN) framework using feed-forward deep neural networks (DNNs) is employed to solve the coupled nonlinear DBF equations solely from the governing equations and boundary conditions, without the need for external simulation or experimental data. Results indicate that lower anisotropic permeability ratios enhance both axial and transverse velocity components, whereas increasing the Darcy number substantially reduces pressure drop due to weakened porous resistance. The proposed framework provides an effective approach for modelling anisotropic porous-medium flows relevant to subsurface transport, filtration, biomedical devices and energy-storage systems.

We present a new kinetic model and its lattice Boltzmann realisation for the simulation of compressible, non-ideal fluid flows. The method employs first-neighbour lattices and introduces a consistent set of correction terms constructed via quasi-equilibrium attractors, ensuring positive-definite and Galilean-invariant Navier–Stokes dissipation rates. This construction circumvents the need for extended stencils or ad hoc regularisation, while maintaining numerical stability and thermodynamic consistency across a broad range of flow regimes. The resulting model accurately reproduces both the Euler and Navier–Stokes hydrodynamic limits. As a stringent validation, we demonstrate, for the first time within a lattice Boltzmann framework, quantitatively accurate simulations of shock–drop interactions at Mach numbers up to 1.47. The proposed approach thus extends the applicability of lattice Boltzmann methods to high-speed, non-ideal compressible flows with a minimal kinetic stencil.

Obtaining predictive low-order models is a central challenge in fluid dynamics. Data-driven frameworks have been widely used to obtain low-order models of aerodynamic systems; yet, resulting models tend to yield predictions that grow unbounded with time. Recently introduced stability-promoting methods can facilitate the identification of bounded models, but tend to require extensive brute-force tuning even in the context of simple academic systems. Here, we show how recent theoretical advances in the long-term boundedness of dynamical systems can be integrated into data-driven modelling frameworks to ensure that resulting models will yield bounded predictions of incompressible flows. Specifically, we propose to solve a particular set of convex semidefinite programming problems to (i) certify whether a system admits a globally attracting bounded set for the chosen modelling parameters, and (ii) compute a model with the optimal (tightest) bound on this globally attracting set. We demonstrate the approach via integration within the sparse identification of nonlinear dynamics (SINDy) modelling framework. Application on two low-order benchmark problems establishes the merits of the approach. We then apply our approach to obtain a low-order (six-mode) model of unsteady separation over a NACA-65(1)-412 aerofoil at ${\textit{Re}}=20\,000$ – a flow that has been notoriously difficult to model using data-driven methods. The resulting model is found to accurately predict the dynamics of unsteady separation, with model predictions remaining bounded indefinitely. We anticipate this work will benefit future efforts in modelling strongly nonlinear flows, especially in settings where physically viable long-term forecasts are paramount.

Numerical solutions of the Navier–Stokes equations in the problem of subsonic jets obtained using a scheme with the 24th-order multi-operators-based dissipation-free approximations are presented and analysed. The spectral properties of the scheme obtained as a result of optimisation for high resolution of small solutions scales are described. The role of added multi-operators in excitation of instability, which make a 23rd-order contribution to the approximation error and form a dissipative component of the scheme, is studied. It is shown how well-known structures of jets with laminar regions of instability development, coherent structures and turbulent regimes at low and moderately high Reynolds numbers are displayed in numerical solutions. Comparisons with experimental data and results of calculations based on large-eddy simulations are given. The results of statistical processing with the calculation of autocorrelations are presented. In particular, their changes with distance from the nozzle are given, characterising different flow regimes at the considered Reynolds numbers. Comparison of spatial and temporal autocorrelations in the region of the turbulent regime indicate good fulfilment of Taylor’s hypothesis on frozen turbulence.

Transient growth analysis has been extensively studied in asymptotically stable flows to identify their short-term amplification of perturbations. Generally, in global transient growth analyses, matrix-free methods are adopted, requiring the construction of adjoint equations, either in the discrete or continuous form. This paper introduces a data-driven algorithm that circumvents the adjoint equations by extracting the optimal initial perturbation and its energy growth over a specified time horizon from transient snapshots of perturbations. This method is validated using data from the linearised complex Ginzburg–Landau equation, backward-facing step flow and the Batchelor vortex. Unlike model-based methods, which require $S$ sets of integrations of the linearised governing equation and its adjoint for $S$ time horizons, the proposed approach collects the snapshots of $S$ time horizons in one integration of the linearised equation. Furthermore, this study provides a robust framework for utilising proper orthogonal decomposition modes to synthesise optimal modes. The developed capacity to conduct transient growth analyses without solving the adjoint equations is expected to significantly reduce the barriers to transient dynamics research.

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.

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.

We present a semi-analytic investigation of the resolvent operator, and its associated forcing and response modes for quasi-one-dimensional shock-laden flows. Using a Green’s function approach, we derive resolvent solutions for isentropic (subsonic and supersonic) and transonic flows with shocks in converging–diverging nozzles of arbitrary geometry. Our analysis demonstrates that shock-induced heightened sensitivity in the resolvent across flow discontinuities leads to significant discrepancies between numerically computed and the analytical input and output modes if shock effects are not properly accounted for. In particular, we find that the resolvent operator exhibits singular behaviour at the shock location. Specifically, the inviscid (where the shock is treated purely as a flow discontinuity) and viscous analytical leading resolvent modes do not converge as the viscosity parameter $\mu \rightarrow 0$, which affects the accuracy of flow control and stability analyses that rely on resolvent-based methods. Furthermore, the derived solutions serve as benchmarks for verifying numerical schemes designed to compute adjoint and resolvent modes in shock-laden flows, ensuring that they capture the correct physical behaviour in the presence of shocks.

The integration of Artificial Intelligence (AI) into computational science (CS) and computational fluid dynamics (CFD) has raised profound epistemological debates concerning the nature of knowledge and its effectiveness in science. A central question in this discourse is whether AI can rival, or potentially surpass, the effectiveness of traditional mathematical methods in addressing the intricate challenges of CFD. In this work, I examine the concept of effectiveness within this context, highlighting the fundamental epistemological distinctions between AI-driven approaches and classical mathematical techniques. First, this analysis identifies four foundational pillars of effectiveness (PoEs) in scientific methods: (i) symmetries, which impose internal structure and coherence; (ii) scale separation, allowing specific treatments for the different scales and their interactions; (iii) sparsity, which simplifies complexity and enhances explicability; and (iv) semantic significance, which fosters abstraction, reasoning and interpretability. Yet, unlike mathematics where rigour ensures credibility by default, AI methods raise additional concerns of robustness and trust. Therefore, beyond the four PoEs, I also discuss credibility as a complementary pillar essential for the adoption of AI in the CFD community. The next critical step is to assess whether, and to what extent, AI can emulate or even outperform the roles and functions traditionally fulfilled by mathematical models. I therefore systematically review if, and how, these four pillar of effectiveness can be applied to AI-based algorithms. I show that those pillars are actually declined in a succession of technical advances that have shown promising results when using AI in CFD.

A data-driven algorithm is proposed that employs sparse data from velocity and/or scalar sensors to forecast the future evolution of three-dimensional turbulent flows. The algorithm combines time-delayed embedding together with Koopman theory and linear optimal estimation theory. It consists of three steps: dimensionality reduction, currently with proper orthogonal decomposition (POD); construction of a linear dynamical system for current and future POD coefficients; and system closure using sparse sensor measurements. In essence, the algorithm establishes a mapping from current sparse data to the future state of the dominant structures of the flow over a specified time window. The method is scalable (i.e. applicable to very large systems), physically interpretable and provides sequential forecasting on a sliding time window of prespecified length. It is applied to the turbulent recirculating flow over a surface-mounted cube (with more than $10^8$ degrees of freedom) and is able to forecast accurately the future evolution of the most dominant structures over a time window at least two orders of magnitude larger that the (estimated) Lyapunov time scale of the flow. Most importantly, increasing the size of the forecasting window only slightly reduces the accuracy of the estimated future states.

Dunn and colleagues (2008) describe and exemplify the use of sophisticated analyses of abstract structural features to reconstruct language histories. The techniques that they use do show some clustering in the groups of languages that they examine; Dunn et al. state that they ‘tend to favor a phylogenetic origin for the signal of relatedness’ (p. 748), and that the results of their test case ‘show a close degree of correspondence to the existing linguistic classification based on sound-meaning correspondences’ (p. 747). We argue that a more parsimonious explanation for the results obtained by Dunn et al.'s methodology is that it accurately maps linguistic geography, the network of contact and diffusion that postdates a proto-language, in most cases corresponding to geographic distance.

Reconstructing near-wall turbulence from wall-based measurements is a critical yet inherently ill-posed problem in wall-bounded flows, where limited sensing and spatially heterogeneous flow–wall coupling challenge deterministic estimation strategies. To address this, we introduce a novel generative modelling framework based on conditional flow matching for synthesising instantaneous velocity fluctuation fields from wall observations, with explicit quantification of predictive uncertainty. Our method integrates continuous-time flow matching with a probabilistic forward operator trained using stochastic weight-averaging Gaussian, enabling zero-shot conditional generation without model re-training. We demonstrate that the proposed approach not only recovers physically realistic, statistically consistent turbulence structures across the near-wall region but also effectively adapts to various sensor configurations, including sparse, incomplete and low-resolution wall measurements. The model achieves robust uncertainty-aware reconstruction, preserving flow intermittency and structure even under significantly degraded observability. Compared with classical linear stochastic estimation and deterministic convolutional neural network methods, our stochastic generative learning framework exhibits superior generalisation for unseen realisations under same flow conditions and resilience under measurement sparsity with quantified uncertainty. This work establishes a robust semi-supervised generative modelling paradigm for data-consistent flow reconstruction and lays the foundation for uncertainty-aware, sensor-driven modelling of wall-bounded turbulence.

The interaction between marine floating structures and projectiles during water entry plays a crucial role in understanding fluid–structure interactions in polar and offshore environments. This study investigates the impact dynamics of a projectile on a floating structure, emphasising the fluid–structure coupling effects, including the impact-induced cavity evolution, stress wave propagation and fragmentation processes. The computational approach integrates fluid dynamics and discrete element methods (CFD-DEM), allowing for detailed simulation of multi-phase interactions during projectile impact. To address the disparity between fluid grid resolution and particle scale, a dual-grid strategy is incorporated, enabling accurate resolution of multi-scale interactions. The results highlight the fundamental mechanisms of impact water entry, where stress waves radiate through the structure, causing local damage and initiating the formation of fragments. These fragments, in turn, influence the stability of the cavity interface and modify the impact dynamics. The interplay between the floating structure’s buoyant support and the surrounding water contributes to complex load variations on the projectile. Ultimately, the study provides insights into the multi-scale fracture mechanisms induced by projectile impact, with potential applications in improving the design and resilience of structures in dynamic marine environments.

Political science is a field rich in multimodal information sources, from televised debates to parliamentary briefings. This paper bridges a gap between computer and political science in multimodal data analysis using audio. The adoption of multimodal analyses in political science (e.g., video/audio with text-as-data approaches) has been relatively slow due to unequal distribution of computational power and skills needed. We provide solutions to challenges encountered when analyzing audio, advancing the potential for multimodal data analysis in political science. Using a dataset of all televised U.S. presidential debates from 1960 to 2020, we focus on three features encountered when analyzing audio data: low-level descriptors (LLDs), such as pitch or energy; Mel-frequency cepstral coefficients (MFCCs); and audio embeddings/encodings, like Wav2Vec. We showcase four applications: (a) forced alignment of audio text using MFCCs, time-stamping transcripts, and speaker information; (b) speech characterization using LLDs; (c) custom-made classification models with audio embeddings and MFCCs; and (d) emotional recognition models using Wav2Vec for classification of discrete emotions and their valence-arousal dominance. We provide explanations to help understand how these features can be applied for different political research questions and advice on vigilance to naive interpretation, for both experienced researchers and those who want to start working with audio.

A linear theory for unsteady aerodynamic effects of the actuator line method (ALM) is developed. This theory is validated using two-dimensional ALM simulations, where we compute the unsteady lift generated by the plunging and pitching motion of a thin aerofoil in uniform flow, comparing the results with Theodorsen’s theory. This comparison elucidates the underlying characteristics and limitations of ALM when applied to unsteady aerodynamics. Numerical simulations were conducted across a range of chord lengths and oscillation frequencies. Comparison of ALM results with theoretical predictions shows consistent accuracy, with all Gaussian parameter choices yielding accurate results at low reduced frequencies. Furthermore, the study indicates that selecting a width parameter ratio of $\varepsilon /c$ (the Gaussian width parameter over the chord length) between 0.33 and 0.4 in ALM yields the closest alignment with analytical results across a broader frequency range. Additionally, a proper definition of angle of attack for a pitching aerofoil is shown to be important for accurate computations. These findings offer valuable guidance for the application of ALM in unsteady aerodynamics and aeroelasticity.