This paper investigates the weakly nonlinear isotropic bidirectional Benney–Luke (BL) equation, which is used to describe oceanic surface and internal waves in shallow water, with a particular focus on soliton dynamics. Using the Whitham modulation theory, we derive the modulation equations associated with the BL equation that describe the evolution of soliton amplitude and slope. By analysing rarefaction waves and shock waves within these modulation equations, we derive the Riemann invariants and modified Rankine–Hugoniot conditions. These expressions help characterise the Mach expansion and Mach reflection phenomena of bent and reverse bent solitons. We also derive analytical formulae for the critical angle and the Mach stem amplitude, showing that as the soliton speed is in the vicinity of unity, the results from the BL equation align closely with those of the Kadomtsev–Petviashvili (KP) equation. Corresponding numerical results are obtained and show excellent agreement with theoretical predictions. Furthermore, as a far-field approximation for the forced BL equation – which models wave and flow interactions with local topography – the modulation equations yield a slowly varying similarity solution. This solution indicates that the precursor wavefronts created by topography moving at subcritical or critical speeds take the shape of a circular arc, in contrast to the parabolic wavefronts observed in the forced KP equation.

The recently proposed near-wall turbulence predictive model quantifies the degree of the superposition and the amplitude modulation exerted by large-scale coherent structures on small scales in the linear and nonlinear terms of the formula, respectively, and achieves the prediction of streamwise velocity in the inner region. However, the multiscale effect and the time shift confirmed in the amplitude modulation have not yet been simultaneously taken into account in the model, which could limit the prediction accuracy especially at high Reynolds numbers. In this study, the role of the nonlinear term in the model is clarified based on high-quality flow data obtained in atmospheric surface layers: it redistributes the energy of the universal signal in the time domain and determines the accuracy of the predictive odd moments. An analysis of the multiscale effect and the time shifts in the nonlinear term is subsequently conducted, followed by a demonstration of the refinement in the quality of the universal signal after separately incorporating them into the model. The amplitude modulation is revealed when the two factors are simultaneously considered, and profiles of the scales that dominate the modulation and time shifts with height is provided. Thus, the nonlinear term of the existing model is modified, proposing an polished scheme that can quantify the nonlinear modulation terms more accurately.

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.

Miscible Rayleigh–Taylor (RT) turbulence exhibits a wide range of length scales in both the velocity and density fields, leading to complex deformations of isoscalar surfaces and enhanced mixing due to nonlinear interactions among different scales. Through high-resolution numerical simulations and a coarse-graining analysis, we demonstrate that the variance of the heavy fluid concentration, initially maximised by the unstable stratification, progressively cascades from larger to smaller scales, eventually dissipates at the smallest scale. The transfer of scalar variance, $\Pi ^Y$, primarily governed by the filtered strain rate tensor, is effectively captured by a nonlinear model that links $\Pi ^Y$ to the isoscalar surface stretching. On the other hand, the backscatter of scalar variance transfer, represented by the negative component of $\Pi ^Y$, is influenced by the filtered vorticity field. Furthermore, we examine the directional anisotropy of scalar transfer in RT turbulence, enhancing the accuracy of the nonlinear model by separating the horizontal mean of the mass fraction from its fluctuating part.

The dispersion behaviour of solutes in flow is crucial to the design of chemical separation systems and microfluidics devices. These systems often rely on coupled electroosmotic and pressure-driven flows to transport and separate chemical species, making the transient dispersive behaviour of solutes highly relevant. However, previous studies of Taylor dispersion in coupled electroosmotic and pressure-driven flows focused on the long-term dispersive behaviour and the associated analyses cannot capture the transient behaviour of solute. Further, the radial distribution of solute has not been analysed. In the current study, we analyse the Taylor dispersion for coupled electroosmotic and pressure-driven flows across all time regimes, assuming a low zeta potential (electric potential at the shear plane), the Debye–Hückel approximation and a finite electric double layer thickness. We first derive analytical expressions for the effective dispersion coefficient in the long-time regime. We also derive an unsteady, two-dimensional (radial and axial) solute concentration field applicable in the latter regime. We next apply Aris’ method of moments to characterise the unsteady propagation of the mean axial position and the unsteady growth of the variance of the solute zone in all time regimes. We benchmark our predictions with Brownian dynamics simulations across a wide and relevant dynamical regime, including various time scales. Lastly, we derive expressions for the optimal relative magnitudes of electroosmotic versus pressure-driven flow and the optimum Péclet number to minimise dispersion across all time scales. These findings offer valuable insights for the design of chemical separation systems, including the optimisation of capillary electrophoresis devices and electrokinetic microchannels and nanochannels.

This study presents an automatic differentiation (AD)-based optimisation framework for flow control in compressible turbulent channel flows. Using a differentiable solver, JAX-Fluids, we designed fully differentiable boundary conditions that allow for the precise calculation of gradients with respect to boundary control variables. This facilitates the efficient optimisation of flow control methods. The framework’s adaptability and effectiveness are demonstrated using two boundary conditions: opposition control and tunable permeable walls. Various optimisation targets are evaluated, including wall friction and turbulent kinetic energy (TKE), across different time horizons. In each optimisation, there were around $4\times 10^4$ control variables and $3\times 10^{9}$ state variables in a single episode. Results indicate that TKE targeted opposition control achieves a more stable and significant reduction in drag, with effective suppression of turbulence throughout the channel. In contrast, strategies that focus directly on minimising wall friction were found to be less effective, exhibiting instability and increased turbulence in the outer region. The tunable permeable walls also show potential to achieve stable drag reduction through a ‘flux-inducing’ mechanism. This study demonstrates the advantages of AD-based optimisation in complex flow control scenarios and provides physical insight into the choice of the quantity of interest for improved optimisation performance.

We perform a comprehensive linear non-modal stability analysis of the Rayleigh–Bénard convection with and without a Poiseuille/Couette flow in Oldroyd-B fluids. In the absence of shear flow, unlike the Newtonian case in which the perturbation energy decays monotonically with time, the interaction between temperature gradient and polymeric stresses can surprisingly cause a transient growth up to 104. This transient growth is maximized at the Hopf bifurcation when the stationary instability dominant in the weakly elastic regime transitions to the oscillatory instability dominant in the strongly elastic regime. In the presence of a Poiseuille/Couette flow, the streamwise-uniform disturbances may achieve the greatest energy amplification, and similar to the pure bounded shear flows, Gmax ∝ Re2 and tmax ∝ Re, where Gmax is the maximum energy growth, tmax the time to attain Gmax, Re the Reynolds number. It is noteworthy that there exist two peaks during the transient energy growth at high-Re cases. Different from the first one which is less affected by the temperature gradient and elasticity, the second peak, at which the disturbance energy is the largest, is simultaneously determined by the temperature gradient, elasticity and shear intensity. Specifically, the polymeric stresses field absorbs energy from the temperature field and base flow, which is partially transferred into the perturbed hydrodynamic field eventually, driving the transient amplification of the perturbed wall-normal vorticity.

Feigenbaum universality is shown to occur in subcritical shear flows. Our testing ground is the counter-rotation regime of the Taylor–Couette flow, where numerical calculations are performed within a small periodic domain. The accurate computation of up to the seventh period-doubling bifurcation, assisted by a purposely defined Poincaré section, has enabled us to reproduce the two Feigenbaum universal constants with unprecedented accuracy in a fluid flow problem. We have further devised a method to predict the bifurcation diagram up to the accumulation point of the cascade based on the detailed inspection of just the first few period-doubling bifurcations. Remarkably, the method is applicable beyond the accumulation point, with predictions remaining valid, in a statistical sense, for the chaotic dynamics that follows.

The influence of parametric forcing on a viscoelastic fluid layer, in both gravitationally stable and unstable configurations, is investigated via linear stability analysis. When such a layer is vertically oscillated beyond a threshold amplitude, large interface deflections are caused by Faraday instability. Viscosity and elasticity affect the damping rate of momentary disturbances with arbitrary wavelength, thereby altering the threshold and temporal response of this instability. In gravitationally stable configurations, calculations show that increased elasticity can either stabilize or destabilize the viscoelastic system. In weakly elastic liquids, higher elasticity increases damping, raising the threshold for Faraday instability, whereas the opposite is observed in strongly elastic liquids. While oscillatory instability occurs in Newtonian fluids for all gravity levels, we find that parametric forcing below a critical frequency will cause a monotonic instability for viscoelastic systems at microgravity. Importantly, in gravitationally unstable configurations, parametric forcing above this frequency stabilizes viscoelastic fluids, until the occurrence of a second critical frequency. This result contrasts with the case of Newtonian liquids, where under the same conditions, forcing stabilizes a system for all frequencies below a single critical frequency. Analytical expressions are obtained under the assumption of long wavelength disturbances predicting the damping rate of momentary disturbances as well as the range of parameters that lead to a monotonic response under parametric forcing.

We explore the instability and oscillation dynamics of barrel-shaped droplets on cylindrical fibres, contributing to a deeper understanding of fibre–droplet interactions critical to both natural systems and industrial applications. Unlike sessile droplets on flat surfaces, droplets on fibres exhibit unique behaviours due to the curvature of the fibre, such as transitions from axisymmetric (barrel) to non-axisymmetric (clamshell) shapes governed by droplet volume, contact angle and fibre radius. Using a linear inviscid theory, we compute the frequency spectrum of barrel-shaped droplets and identify stability thresholds for the barrel-to-clamshell transition by examining the first rocking mode, with a focus on the role of contact line conditions. This analysis resolves experimental anomalies concerning the stability of half-barrel-shaped droplets on hydrophobic fibres. Our findings also reveals diverse frequency spectra: droplets on thin fibres exhibit Rayleigh–Lamb-like spectral features, while those on thicker fibres show reduced sensitivity to azimuthal wavenumber. Interestingly, the instability of sectoral modes on thick fibres resembles the Rayleigh–Plateau instability of static rivulets, with fibre curvature slightly reducing growth rates at small axial wavenumbers but increasing them at larger ones.

Being thicker and lighter than the oceanic lithosphere, the continental lithosphere exerts a thermal blanket effect on the convective mantle by locally accumulating heat and altering the flow structure, which in turn affects continent motion. This thermal–mechanic feedback has been studied through a simplified model of a thermally insulating plate floating over a bottom-heated convective fluid, which shows that plate mobility enhances with plate size and a unidirectionally moving mode (UMM) emerges for sufficiently large plates. Nevertheless, apart from bottom heating, the mantle is also subject to internal heating induced by radioactive decay. How the addition of internal heating affects the dynamic coupling is still unclear, which motivates the present study. Numerical simulation results show that the effect varies with plate size. For small plates, as internal heating intensifies, plate motion becomes increasingly persistent and the critical plate size for the UMM decreases. This results from the enhanced thermal blanket effect under intensified internal heating, which enables a faster generation of hot plumes to boost plate motion during its slowdown. Most notably, the addition of internal heating brings a new mode for large plates – a permanently stagnant mode (PSM) – in which the plate oscillates permanently above a hot up-welling with down-wellings locating far away. The critical size for the PSM decreases as internal heating intensifies. In the PSM, the symmetry between cold and hot plumes breaks. Implications of these findings for the dynamic coupling on Earth and Mars are discussed.

A new lattice Boltzmann model (LBM) is presented to describe chemically reacting multicomponent fluid flow in homogenised porous media. In this work, towards further generalising the multicomponent reactive lattice Boltzmann model, we propose a formulation which is capable of performing reactive multicomponent flow computation in porous media at the representative elementary volume (REV) scale. To that end, the submodel responsible for interspecies diffusion has been upgraded to include Knudsen diffusion, whereas the kinetic equations for the species, the momentum and the energy have been rewritten to accommodate the effects of volume fraction of porous media through careful choice of the equilibrium distribution functions. Verification of the mesoscale kinetic system of equations by a Chapman–Enskog analysis reveals that at the macroscopic scale, the homogenised Navier–Stokes equations for compressible multicomponent reactive flows are recovered. The dusty gas model (DGM) capability hence formulated is validated over a wide pressure range by comparison of experimental flow rates of component species counter diffusing through capillary tubes. Next, for developing a capability to compute heterogeneous reactions, source terms for maintaining energy and mass balance across the fluid phase species and the surface adsorbed phase species are proposed. The complete model is then used to perform detailed chemistry simulations in porous electrodes of a solid oxide fuel cell (SOFC), thereby predicting polarisation curves which are of practical interest.

The effect of nucleation on cavitation inception in a high-Reynolds-number von Kármán wake from a bluff two-dimensional hydrofoil is studied experimentally in a variable pressure water tunnel. Nucleation effects are studied by seeding the flow with sparse monodisperse nuclei populations, with the critical pressure nominally equal to vapour pressure. The injected nuclei population and incipient cavitation events were imaged simultaneously using high-speed cameras to precisely quantify the number of activated nuclei of the total available. Three-dimensional spatial characterisation (orientation and location) of the incipient structures is obtained using two high-speed cameras mounted to the side and below the tunnel test section. Inception was observed predominantly in the stretched cores of secondary structures, with a negligible proportion of events occurring in the primary vortices. A broad peak in the vertical angle distribution is observed about the streamwise axis; however, events at all angles are seen. A symmetric distribution was observed for the horizontal angle, with a dominant orientation $45^{\circ }$ from the free-stream direction. The majority of events occur at approximately one hydrofoil thickness downstream of the hydrofoil trailing edge, with a bimodal symmetric distribution about the hydrofoil vertical centre plane. Nuclei activation rate is determined from the acoustic measurements, and was found to be proportional to the number of the injected nuclei. A power law increase in activation rate was observed following a decrease in cavitation number and an increase in Reynolds number. The nuclei activation rate was of the order of $0.1{-}10 \, \mathrm {s^{-1}}$, which combined with seeding rates of the orderof $100{-}1000 \, \mathrm {s^{-1}}$ reveals inception to be a rare occurrence (0.001 %–10 % of nuclei being activated), requiring the confluence of two unlikely events, the occurrence of a subvapour pressure vortex core with capture of a sufficiently weak nuclei. The presented study provides new insights into the physics of cavitation nucleation and inception and provides a comprehensive dataset for development of computational models.

We experimentally identify a rotational motion of a single microalga (Chlamydomonas reinhardtii) within a microcontainer believed to be induced by one defective flagellum. We numerically adapt the classic two-dimensional squirmer model to replicate this unique motion by partially inhibiting the slip velocity on the boundaries of the squirmer. Subsequently, we employ a lattice Boltzmann method to simulate the motion of the single microalga with one defective flagellum. We examine the influence of swimming Reynolds numbers, self-propelling strength ($\beta$) and angle ($\alpha$) on the locomotion of the squirmer with one defective flagellum. The results indicate that a large $\beta$ leads to a large rotational diameter, positively correlating with the speed. Additionally, we observe that a low self-propelling strength ($\beta =0.5$) yields a monotonically increasing speed for the squirmer with $\alpha$. In general, high $\beta$ values result in fast speeds for the squirmer. This differs from the behaviour observed in a classic squirmer ($\alpha =360^{\circ }$), where high $\beta$ leads to a slow speed of puller ($\beta \gt 0$) owing to weak fluid inertia effects. Meanwhile, the energy expenditure increases monotonically with $\alpha$, contrasting with the non-monotonic trends observed for swimming speed and rotational diameter.

We solve a Bayesian inverse Navier–Stokes (N–S) problem that assimilates velocimetry data by jointly reconstructing a flow field and learning its unknown N–S parameters. We devise an algorithm that learns the most likely parameters of a Carreau shear-thinning viscosity model, and estimates their uncertainties, from velocimetry data of a shear-thinning fluid. We conduct a magnetic resonance velocimetry experiment to obtain velocimetry data of an axisymmetric laminar jet in an idealised medical device (US Food and Drug Administration’s benchmark nozzle) for a blood analogue fluid. The algorithm successfully reconstructs the flow field and learns the most likely Carreau parameters. Predictions from the learned model agree well with rheometry measurements. The algorithm accepts any differentiable algebraic viscosity model, and can be extended to more complicated non-Newtonian fluids (e.g. Oldroyd-B fluid if a viscoelastic model is incorporated).

A linear stability analysis of a soluble surfactant-laden liquid film flowing down a compliant substrate is performed. Our purpose is to expand the prior studies (Carpenter and Garrad 1985 J. Fluid Mech. 155, 465–510; Alexander et al., 2020 J. Fluid Mech. 900, A40) by incorporating a soluble surfactant into the flow configuration. As a result, we formulate the Orr–Sommerfeld-type boundary value problem and solve it analytically by using the long-wave series expansion as well as numerically by using the Chebyshev spectral collocation method in an arbitrary wavenumber regime for infinitesimal disturbances. The long-wave result reveals that surface instability is stabilized in the presence of a surfactant, whereas it is destabilized in the presence of a compliant substrate. These opposing impacts suggest an analytical relationship between parameters associated with the soluble surfactant and compliant wall, ensuring the same critical Reynolds number for the emergence of surface instability corresponding to both surfactant-laden film flow over a compliant wall and surfactant-free film flow over a non-compliant wall. In the arbitrary wavenumber regime, along with the surface mode, we identify two additional modes based on their distinct phase speeds. Specifically, the wall mode emerges in the finite wavenumber regime, while the shear mode emerges only when the Reynolds number is large. As the surfactant Marangoni number increases, the wall mode destabilizes, resulting in a different outcome from the surface mode. Moreover, increasing the value of the ratio of adsorption and desorption rate constants stabilizes surface instability but destabilizes wall mode instability. As a result, we perceive that the soluble surfactant-laden film flow is linearly more unstable than the insoluble one due to surface instability but linearly more stable than the insoluble one due to wall mode instability. Additionally, we see a peculiar behaviour of base surface surfactant concentration on the primary instability. In fact, it has a specific value depending on adsorption and desorption rate constants below which surface instability stabilizes but wall mode instability destabilizes, whereas above which an opposite phenomenon occurs. Finally, in the high-Reynolds-number regime, we can suppress shear mode instability by raising the surfactant Marangoni number and the ratio of adsorption and desorption rate constants when the angle of inclination is sufficiently small. Unlike surface instability, the base surface surfactant concentration exhibits both stabilizing and destabilizing influences on shear mode instability.

Spontaneous flow reversals in buoyancy-driven flows are ubiquitous in many fields of science and engineering, often characterized by violent, intermittent occurrences. In this study, we present a complex-network-based reduced-order model to analyse intermittent events in turbulent flows, using temporal and spatial snapshot data. This framework combines elements of dynamical system theory with network science. We demonstrate its utility by applying it to data sequences from intermittent flow reversal events in two-dimensional thermal convection. This approach has proven robust in detecting and quantifying structures and predicting reversals. Additionally, it provides a perspective on the physical mechanisms underlying flow reversals through cluster evolution. This purely data-driven methodology shows the potential to enhance our understanding, prediction and control of turbulent flows and complex systems.

The wake systems of ducted and conventional marine propellers are compared for a highly loaded condition by exploiting results of large eddy simulations, conducted on a cylindrical grid consisting of 3.5 billion points. The results demonstrate a dramatic change of both performance and flow physics, due to the nozzle. The efficiency of propulsion is increased by about $30\,\%$, but the thrust generated by the propeller is reduced, replaced in most part by that produced by its nozzle. As a result, weaker coherent structures are shed in the wake on the ducted propeller, compared with the conventional one. Meanwhile, the tip leakage vortices experience a faster instability into smaller turbulent structures. Therefore, the wake signature of the ducted propeller, detrimental to its interaction with downstream bodies, is reduced, compared with that of the conventional propeller operating with no duct. The source of the faster instability of the tip leakage vortices is different from the typical one of the tip vortices shed by conventional propellers. The latter is attributable to phenomena of short- and long-wave instabilities of the helices of each tip vortex, eventually leading to mutual inductance, leapfrogging and breakup into turbulence. In contrast, the former is tied to the shear developed between the tip leakage vortices and the boundary layer of the inner surface of the nozzle, rather than to the interaction between vortices shed by different blades.

We investigate through numerical simulations the hydrodynamic interactions between two rigid spherical particles suspended on the axis of a cylindrical tube filled with an elastoviscoplastic fluid subjected to pressure-driven flow. The simulations are performed by the finite-element method with the arbitrary Lagrangian–Eulerian formulation. We carry out a parametric analysis to examine the impact of the yield stress and relaxation time of the fluid and of particle confinement on the dynamics of the system. We identify master curves of the particle relative velocity as a function of the inter-particle distance. When the yield stress of the suspending phase is much lower than the viscous stress, those curves highlight short-range attractive interactions and long-range repulsive interactions between particles, with the latter specifically promoting their alignment. As the yield stress increases, the attractive interaction is replaced by stasis at short distance, characterised by a vanishing relative velocity and the formation of an unyielded region that connects the two spheres, where the fluid behaves like a viscoelastic solid. Additionally, the combined effects of plasticity and elasticity enhance the repulsion between the particles, promoting their ordering. Also, increasing the confinement of the particles enhances repulsion, thus allowing us to achieve ordering within shorter lengths in the flow direction. Reducing shear thinning amplifies peak relative velocities and expands the attractive region due to increased viscoelastic stresses and stress gradients. While a stable equilibrium may appear at larger separations, its impact is limited by low relative velocities.

Direct numerical simulations of the injection of a pulsed round liquid jet in a stagnant gas are performed in a series of runs of geometrically progressing resolution. The Reynolds and Weber numbers and the density ratio are sufficiently large for reaching a complex high-speed atomisation regime but not so large so that the small length scales of the flow are impossible to resolve, except for a very small liquid-sheet thickness. The Weber number based on grid size is then small, an indication that the simulations are very well resolved. Computations are performed using octree adaptive mesh refinement with a finite volume method and height-function computation of curvature, down to a specified minimum grid size $\varDelta$. Qualitative analysis of the flow and its topology reveals a complex structure of ligaments, sheets, droplets and bubbles that evolve and interact through impacts, ligament breakup, sheet rupture and engulfment of air bubbles in the liquid. A rich gallery of images of entangled structures is produced. Most processes occurring in this type of atomisation are reproduced in detail, except at the instant of thin sheet perforation or breakup. We analyse droplet statistics, showing that as the grid resolution is increased, the small-scale part of the distribution does not converge, and contains a large number of droplets close in order of magnitude to the minimum grid size with a significant peak at $d = 3\varDelta$. This non-convergence arises from the numerical sheet breakup effect, in which the interface becomes rough just before it breaks. The rough appearance of the interface is associated with a high-wavenumber oscillation of the curvature. To recover convergence, we apply the controlled ‘manifold death’ numerical procedure, in which thin sheets are detected, and then pierced by fiat before they reach a set critical thickness $h_c$ that is always larger than $6 \varDelta$. This allows convergence of the droplet frequency above a certain critical diameter $d_c$, above and close to $h_c$. A unimodal distribution is observed in the converged range. The number of holes pierced in the sheet is a free parameter in the manifold death procedure; however, we use the Kibble–Zurek theory to predict the number of holes expected on heuristic physical grounds.