Invariant maps are a useful tool for turbulence modelling, and the rapid growth of machine learning-based turbulence modelling research has led to renewed interest in them. They allow different turbulent states to be visualised in an interpretable manner and provide a mathematical framework to analyse or enforce realisability. Current invariant maps, however, are limited in machine learning models by the need for costly coordinate transformations and eigendecomposition at each point in the flow field. This paper introduces a new polar invariant map based on an angle that parametrises the relationship of the principal anisotropic stresses, and a scalar that describes the anisotropy magnitude relative to a maximum value. The polar invariant map reframes realisability in terms of a limiting anisotropy magnitude, allowing for new and simplified approaches to enforcing realisability that do not require coordinate transformations or explicit eigendecomposition. Potential applications to machine learning-based turbulence modelling include post-processing corrections for realisability, realisability-informed training, turbulence models with adaptive coefficients and general tensor basis models. The relationships to other invariant maps are illustrated through examples of plane channel flow and square duct flow. Sample calculations are provided for a comparison with a typical barycentric map-based method for enforcing realisability, showing an average 62 % reduction in calculation time using the equivalent polar formulation. The results provide a foundation for new approaches to enforcing realisability constraints in Reynolds-averaged turbulence modelling.

In recent years, integrating physical constraints within deep neural networks has emerged as an effective approach for expediting direct numerical simulations in two-phase flow. This paper introduces physics-informed neural networks (PINNs) that utilise the phase-field method to model three-dimensional two-phase flows. We present a fully connected neural network architecture with residual blocks and spatial parallel training using the overlapping domain decomposition method across multiple graphics processing units to enhance the accuracy and computational efficiency of PINNs for the phase-field method (PF-PINNs). The proposed PINNs framework is applied to a bubble rising scenario in a three-dimensional infinite water tank to quantitatively assess the performance of PF-PINNs. Furthermore, the computational cost and parallel efficiency of the proposed method was evaluated, demonstrating its potential for widespread application in complex training environments.

The pressure effects on the mixing fields of non-reacting and reacting jets in cross-flow are studied using large eddy simulation (LES). A hydrogen jet diluted with 30 % helium is injected perpendicularly into a cross-stream of air at four different pressures: 1, 4, 7 and 15 bar. The resulting interaction and the mixing fields under non-reacting and reacting conditions are simulated using LES. The subgrid scale combustion is modelled using a revised flamelet model for the partially premixed combustion. Good agreement of computed and measured velocity fields for reacting and non-reacting conditions is observed. Under non-reacting conditions, the mixing field shows no sensitivity to the pressure, whereas notable changes are observed for reacting conditions. The lifted flame at 1 bar moves upstream and attaches to the nozzle as the pressure is increased to 4 bar and remains so for the other elevated pressures because of the increasing burning mass flux with pressure. This attached flame suppresses the fuel–air mixing in the near-nozzle region. The premixed and non-premixed contributions to the overall heat release in the partially premixed combustion are analysed. The non-premixed contribution is generally low and occurs in the near-field region of the fuel jet through fuel-rich mixtures in the shear layer regions, and decreases substantially further with the increase in pressure. Hence, the predominant contributions are observed to come from premixed modes and these contributions increase with pressure.

Several million years of natural evolution have endowed marine animals with high flexibility and mobility. A key factor in this achievement is their ability to modulate stiffness during swimming. However, an unresolved puzzle remains regarding how muscles modulate stiffness, and the implications of this capability for achieving high swimming efficiency. Inspired by this, we proposed a self-propulsor model that employs a parabolic stiffness-tuning strategy, emulating the muscle tensioning observed in biological counterparts. Furthermore, efforts have been directed towards developing the nonlinear vortex sheet method, specifically designed to address nonlinear fluid–structure coupling problems. This work aims to analyse how and why nonlinear tunable stiffness influences swimming performance. Numerical results demonstrate that swimmers with nonlinear tunable stiffness can double their speed and efficiency across nearly the entire frequency range. Additionally, our findings reveal that high-efficiency biomimetic propulsion originates from snap-through instability, which facilitates the emergence of quasi-quadrilateral swimming patterns and enhances vortex strength. Moreover, this study examines the influence of nonlinear stiffness on swimming performance, providing valuable insights into the optimisation of next-generation, high-performance, fish-inspired robotic systems.

We investigate the energy transfer from the mean profile to velocity fluctuations in channel flow by calculating nonlinear optimal disturbances, i.e. the initial condition of a given finite energy that achieves the highest possible energy growth during a given fixed time horizon. It is found that for a large range of time horizons and initial disturbance energies, the nonlinear optimal exhibits streak spacing and amplitude consistent with direct numerical simulation (DNS) at least at ${Re}_\tau = 180$, which suggests that they isolate the relevant physical mechanisms that sustain turbulence. Moreover, the time horizon necessary for a nonlinear disturbance to outperform a linear optimal is consistent with previous DNS-based estimates using eddy turnover time, which offers a new perspective on how some turbulent time scales are determined.

The transient dynamics of a wake vortex, modelled as a strong swirling $q$-vortex, is investigated with a focus on optimal transient growth driven by continuous eigenmodes associated with continuous spectra. The pivotal contribution of viscous critical-layer eigenmodes (Lee and Marcus, J. Fluid Mech. vol. 967, 2023, p. A2) amongst the entire eigenmode families to optimal perturbations is numerically confirmed, using a spectral collocation method for a radially unbounded domain that ensures correct analyticity and far-field behaviour. The consistency of the numerical method across different sensitivity tests supports the reliability of the results and provides flexibility for tuning. Both axisymmetric and helical perturbations with axial wavenumbers of order unity or less are examined through linearised theory and nonlinear simulations, yielding results that align with existing literature on energy growth curves and optimal perturbation structures. The initiation process of transient growth is also explored, highlighting its practical relevance. Inspired by ice crystals in contrails, the backward influence of inertial particles on the vortex flow, particularly through particle drag, is emphasised. In the pursuit of optimal transient growth, particles are initially distributed at the periphery of the vortex core to disturb the flow. Two-way coupled vortex–particle simulations reveal clear evidence of optimal transient growth during ongoing vortex–particle interactions, reinforcing the robustness and significance of transient growth in the original nonlinear vortex system over finite time periods.

In the dynamical systems approach to turbulence, unstable periodic orbits (UPOs) provide valuable insights into system dynamics. Such UPOs are usually found by shooting-based Newton searches, where constructing sufficiently accurate initial guesses is difficult. A common technique for constructing initial guesses involves detecting recurrence events by comparing past and future flow states using their $L_2$-distance. An alternative method uses dynamic mode decomposition (DMD) to generate initial guesses based on dominant frequencies identified from a short time series, which are signatures of a nearby UPO. However, DMD struggles with continuous symmetries. To address this drawback, we combine symmetry-reduced DMD (SRDMD) introduced by Marensi et al. (2023, J. Fluid Mech., vol. 954, A10), with sparsity promotion. This combination provides optimal low-dimensional representations of the given time series as a time-periodic function, allowing any time instant along this function to serve as an initial guess for a Newton solver. We also discuss how multi-shooting methods operate on the reconstructed trajectories, and we extend the method to generate initial guesses for travelling waves. We demonstrate SRDMD as a method complementary to recurrent flow analysis by applying it to data obtained by direct numerical simulations of three-dimensional plane Poiseuille flow at the friction Reynolds number $\textit{Re}_\tau \approx51$ ($\textit{Re}=802$), explicitly taking a continuous shift symmetry in the streamwise direction into account. The resulting unstable relative periodic orbits cover relevant regions of the state space, highlighting their potential for describing the flow.

We present a numerical scheme that solves for the self-similar viscous fingers that emerge from the Saffman–Taylor instability in a divergent wedge. This is based on the formulation by Ben Amar (1991, Phys. Rev. A, vol. 44, pp. 3673–3685). It is demonstrated that there exists a countably infinite set of selected solutions, each with an associated relative finger angle, and furthermore, solutions can be characterised by the number of ripples located at the tip of their finger profiles. Our numerical scheme allows us to observe these ripples and measure them, demonstrating that the amplitudes are exponentially small in terms of the surface tension; the selection mechanism is driven by these exponentially small contributions. A recently published paper derived the selection mechanism for this problem using exponential asymptotic analytical techniques, and obtained bifurcation diagrams that we compare with our numerical results.

This paper presents a peridynamics-based computational approach for modelling coupled fluid flow and heat transfer problems. A new thermo-hydrodynamic peridynamics model is formulated with the semi-Lagrangian scheme and non-local operators. To enhance accuracy and numerical stability, a multi-horizon scheme is developed to introduce distinct horizons for the flow field and thermal field. The multi-horizon scheme helps to capture the convective zone and complex thermal flow pattern while effectively mitigating possible oscillations in temperature. We validate the computational approach using benchmarks and numerical examples including heat conduction, natural convection in a closed cavity, and Rayleigh–Bénard convection cells. The results demonstrate that the proposed method can accurately capture typical thermal flow behaviours and complex convective patterns. This work offers a new foundation for future development of a unified peridynamics framework for robust, comprehensive multi-physics analysis of thermal fluid–solid interaction problems with complex evolving discontinuities in solids.

Geophysical and astrophysical fluid flows are typically driven by buoyancy and strongly constrained at large scales by planetary rotation. Rapidly rotating Rayleigh–Bénard convection (RRRBC) provides a paradigm for experiments and direct numerical simulations (DNS) of such flows, but the accessible parameter space remains restricted to moderately fast rotation rates (Ekman numbers ${ {Ek}} \gtrsim 10^{-8}$), while realistic ${Ek}$ for geo- and astrophysical applications are orders of magnitude smaller. On the other hand, previously derived reduced equations of motion describing the leading-order behaviour in the limit of very rapid rotation ($ {Ek}\to 0$) cannot capture finite rotation effects, and the physically most relevant part of parameter space with small but finite ${Ek}$ has remained elusive. Here, we employ the rescaled rapidly rotating incompressible Navier–Stokes equations (RRRiNSE) – a reformulation of the Navier–Stokes–Boussinesq equations informed by the scalings valid for ${Ek}\to 0$, recently introduced by Julien et al. (2024) – to provide full DNS of RRRBC at unprecedented rotation strengths down to $ {Ek}=10^{-15}$ and below, revealing the disappearance of cyclone–anticyclone asymmetry at previously unattainable Ekman numbers (${Ek}\approx 10^{-9}$). We also identify an overshoot in the heat transport as ${Ek}$ is varied at fixed $\widetilde { {Ra}} \equiv {Ra}{Ek}^{4/3}$, where $Ra$ is the Rayleigh number, associated with dissipation due to ageostrophic motions in the boundary layers. The simulations validate theoretical predictions based on thermal boundary layer theory for RRRBC and show that the solutions of RRRiNSE agree with the reduced equations at very small ${Ek}$. These results represent a first foray into the vast, largely unexplored parameter space of very rapidly rotating convection rendered accessible by RRRiNSE.

We perform the first mapping of the ideological positions of European parties using generative Artificial Intelligence (AI) as a “zero-shot” learner. We ask OpenAI’s Generative Pre-trained Transformer 3.5 (GPT-3.5) to identify the more “right-wing” option across all possible duplets of European parties at a given point in time, solely based on their names and country of origin, and combine this information via a Bradley–Terry model to create an ideological ranking. A cross-validation employing widely-used expert-, manifesto- and poll-based estimates reveals that the ideological scores produced by Large Language Models (LLMs) closely map those obtained through the expert-based evaluation, i.e., CHES. Given the high cost of scaling parties via trained coders, and the scarcity of expert data before the 1990s, our finding that generative AI produces estimates of comparable quality to CHES supports its usage in political science on the grounds of replicability, agility, and affordability.

Rogue waves (RWs) can form on the ocean surface due to the well-known quasi-four-wave resonant interaction or superposition principle. The first is known as the nonlinear focusing mechanism and leads to an increased probability of RWs when unidirectionality and narrowband energy of the wave field are satisfied. This work delves into the dynamics of extreme wave focusing in crossing seas, revealing a distinct type of nonlinear RWs, characterised by a decisive longevity compared with those generated by the dispersive focusing (superposition) mechanism. In fact, through fully nonlinear hydrodynamic numerical simulations, we show that the interactions between two crossing unidirectional wave beams can trigger fully localised and robust development of RWs. These coherent structures, characterised by a typical spectral broadening then spreading in the form of dual bimodality and recurrent wave group focusing, not only defy the weakening expectation of quasi-four-wave resonant interaction in directionally spreading wave fields, but also differ from classical focusing mechanisms already mentioned. This has been determined following a rigorous lifespan-based statistical analysis of extreme wave events in our fully nonlinear simulations. Utilising the coupled nonlinear Schrödinger framework, we also show that such intrinsic focusing dynamics can be captured by weakly nonlinear wave evolution equations. This opens new research avenues for further explorations of these complex and intriguing wave phenomena in hydrodynamics as well as other nonlinear and dispersive multi-wave systems.

The lattice Boltzmann method has become a popular tool for simulating complex flows, including incompressible turbulent flows; however, as an artificial compressibility method, it can generate spurious pressure oscillations whose impact on the statistics of incompressible turbulence has not been systematically examined. In this work, we propose a theoretical approach to analyse the origin of compressibility-induced oscillations (CIOs) and explore ways to suppress or remove them. We begin by decomposing the velocity field and pressure field each into the solenoidal component and the compressive component, and then study the evolution of these two components analytically and numerically. The analysis yields an evolution equation of the mean-square pressure fluctuation which reveals several coupling effects of the two components. The evolution equation suggests that increasing the bulk-to-shear viscosity ratio can suppress CIOs, which is confirmed by numerical simulations. Furthermore, based on the derived evolution equation and data from the simulation, a model is developed to predict the long-term behaviours of the mean-square pressure fluctuations. In the case of decaying turbulence in a periodic domain, we show that the Helmholtz–Hodge decomposition can be used to obtain the solenoidal components reflecting the true evolution of incompressible turbulent flow, from the mesoscopic artificial compressibility approach. The study provides general theoretical guidelines to understand, suppress and even remove CIOs in other related pseudo-compressibility methods.

Understanding the mechanisms behind the remote triggering of landslides by seismic waves at micro-strain amplitude is essential for quantifying seismic hazards. Granular materials provide a relevant model system to investigate landslides within the unjamming transition framework, from solid to liquid states. Furthermore, recent laboratory experiments have revealed that ultrasound-induced granular avalanches can be related to a reduction in the interparticle friction through shear acoustic lubrication of the contacts. However, investigating slip at the scale of grain contacts within an optically opaque granular medium remains a challenging issue. Here, we propose an original coupling model and numerically investigate two-dimensional dense granular flows triggered by basal acoustic waves. We model the triggering dynamics at two separated time scales – one for grain motion (milliseconds) and the other for ultrasound (10 ${\rm \mu} {\rm s}$) – relying on the computation of vibrational modes with a discrete element method through the reduction of the local friction. We show that ultrasound predominantly propagates through the strong-force chains, while the ultrasound-induced decrease of interparticle friction occurs in the weak contact forces perpendicular to the strong-force chains. This interparticle friction reduction initiates local rearrangements at the grain scale that eventually lead to a continuous flow through a percolation process at the macroscopic scale – with a delay depending on the proximity to the failure. Consistent with experiments, we show that ultrasound-induced flow appears more uniform in space than pure gravity-driven flow, indicating the role of an effective temperature by ultrasonic vibration.

An efficient compression scheme for modal flow analysis is proposed and validated on data sequences of compressible flow through a linear turbomachinery blade row. The key feature of the compression scheme is a minimal, user-defined distortion of the mutual distance of any snapshot pair in phase space. Through this imposed feature, the model reduction process preserves the temporal dynamics contained in the data sequence, while still decreasing the spatial complexity. The mathematical foundation of the scheme is the fast Johnson–Lindenstrauss transformation (FJLT) which uses randomized projections and a tree-based spectral transform to accomplish the embedding of a high-dimensional data sequence into a lower-dimensional latent space. The compression scheme is coupled to a proper orthogonal decomposition and dynamic mode decomposition analysis of flow through a linear blade row. The application to a complex flow-field sequence demonstrates the efficacy of the scheme, where compression rates of two orders of magnitude are achieved, while incurring very small relative errors in the dominant temporal dynamics. This FJLT technique should be attractive to a wide range of modal analyses of large-scale and multi-physics fluid motion.

When they occur, azimuthal thermoacoustic oscillations can detrimentally affect the safe operation of gas turbines and aeroengines. We develop a real-time digital twin of azimuthal thermoacoustics of a hydrogen-based annular combustor. The digital twin seamlessly combines two sources of information about the system: (i) a physics-based low-order model; and (ii) raw and sparse experimental data from microphones, which contain both aleatoric noise and turbulent fluctuations. First, we derive a low-order thermoacoustic model for azimuthal instabilities, which is deterministic. Second, we propose a real-time data assimilation framework to infer the acoustic pressure, the physical parameters, and the model bias and measurement shift simultaneously. This is the bias-regularized ensemble Kalman filter, for which we find an analytical solution that solves the optimization problem. Third, we propose a reservoir computer, which infers both the model bias and measurement shift to close the assimilation equations. Fourth, we propose a real-time digital twin of the azimuthal thermoacoustic dynamics of a laboratory hydrogen-based annular combustor for a variety of equivalence ratios. We find that the real-time digital twin (i) autonomously predicts azimuthal dynamics, in contrast to bias-unregularized methods; (ii) uncovers the physical acoustic pressure from the raw data, i.e. it acts as a physics-based filter; (iii) is a time-varying parameter system, which generalizes existing models that have constant parameters, and capture only slow-varying variables. The digital twin generalizes to all equivalence ratios, which bridges the gap of existing models. This work opens new opportunities for real-time digital twinning of multi-physics problems.

Understanding the linear growth of disturbances due to external forcing is crucial for flow stability analysis, flow control, and uncertainty quantification. These applications typically require a large number of forward simulations of the forced linearized dynamics, often in a brute-force fashion. When dealing with simple steady-state or periodic base flows, there exist powerful and cost-effective solution operator techniques. Once these solution operators are constructed, they can be used to determine the response to various forcings with negligible computational cost. However, these methods do not apply to problems with arbitrarily time-dependent base flows. This paper develops and investigates reduced-order modelling with time-dependent bases to build low-rank solution operators for forced linearized dynamics with arbitrarily time-dependent base flows. In particular, we use forced optimally time-dependent decomposition (f-OTD), which extracts the time-dependent correlated structures of the flow response to various excitations. Several demonstrations are included to illustrate the utility of the f-OTD low-rank approximation for performing global transient stability analysis. Additionally, we demonstrate the application of f-OTD in computing the post-transient response of linearized Navier–Stokes equations to a large number of impulses, which has applications in flow control.

Liquid plug formation in thin channels due to the Plateau–Rayleigh instability of a liquid film is observed in a variety of fields. In this paper, complementarity between theoretical solutions and direct numerical simulations (DNS) based on a front-tracking algorithm is explored to evaluate the importance of inertia for the case of a cylindrical capillary. A linear stability analysis is first performed and DNS results are then used to investigate the spatial distributions of inertial, convective and viscous terms of the Navier–Stokes equation. The existence of both viscous and inertial regimes is evidenced with a threshold given by the film thickness. The presence of the core fluid slows down the instability. In the viscous regime, predictions of the lubrication theory are verified. An example of liquid water as the outer fluid film and water vapour as the inner core fluid is simulated with application to the fuel cells.