To address the limitation of the generalised Reynolds analogy (GRA) in handling flows with a spatial mismatch between velocity and temperature extrema, we propose a zonal and regime-based GRA which integrates a zonal decomposition approach based on the extrema of velocity and temperature profiles with a regime-based approach that accounts for different temperature–velocity (T–V) relations. The new GRA is verified using compressible turbulent Couette–Poiseuille (C–P) flow, which occurs between two plane plates driven by the combination of relative moving walls and the application of a pressure gradient. Direct numerical simulations (DNS) are implemented at ${\textit{Re}}_0 = 4000$, $\textit{Ma}_0 = 0.8$ and $1.5$. Two flow regimes are recognised: one is the Couette regime (C regime), featuring opposite-direction wall frictions on the bottom and top walls, and the other is the Poiseuille regime (P regime), characterised by same-direction wall frictions. For C-regime flow, the temperature maximum point and the minimum magnitude point of the velocity gradient divide the entire channel into three zones, with each zone modelled via canonical GRA. For P-regime flow, the velocity maximum point presents a strong singularity for canonical GRA. We propose a new set of T–V relations with non-uniform distribution of the effective Prandtl number (${\textit{Pr}}_e$) rather than the typical constant-${\textit{Pr}}_e$ assumption. Comparisons with DNS results indicate that the new T–V relation improves the prediction of temperature profile in compressible C–P turbulence, particularly for the two P-regime flows with higher $\textit{Ma}_0$, where the original GRA model shows clear deviations from the DNS.

A large-scale parametric study of the flow over the prolate spheroid is presented to understand the effect of Reynolds number and angle of attack on the separation, the wake formation and the loads. Large-eddy simulation is performed for six Reynolds numbers ranging from ${\textit{Re}} = 0.15\times 10^6$ to $4 \times 10^6$ and for eight angles of attack ranging from $\alpha = 10^\circ$ to $\alpha = 90^\circ$. For all the cases considered, the boundary layer separates symmetrically and forms a recirculation region. Several distinct flow topologies are observed that can be grouped into three categories: proto-vortex, coherent vortex and recirculating wake. In the proto-vortex state, the recirculation does not have a distinct centre of rotation, instead, a two-layer detached flow structure is formed. In the coherent vortex state, the separated shear layer rolls into a three-dimensional vortex that is aligned with the axis of the spheroid. This vortex has a clear centre of rotation corresponding to a minimum of pressure and transforms the transverse momentum from the separated shear layer into axial momentum. In the recirculating wake regime, the recirculation is incoherent and the primary separation forms a dissipative shear layer that is convected in the direction of the free stream. This symmetric pair of shear layers bounds a low-momentum recirculating cavity on the leeward side of the spheroid. The properties of these states are not constant, but evolve along the axis of the spheroid and are dictated by the characteristics of the boundary layer at separation. The variation of the flow with Reynolds number and angle of attack is described, and its connection to the loads on the spheroid are discussed.

This paper aims to elucidate the physical mechanisms underlying airfoil–vortex gust interaction and mitigation. The vortex gust mitigation problem consists in finding the pitch rate sequence that minimises the gust-induced lift disturbance of an NACA0012 airfoil at Reynolds number 1000. The instantaneous flow fields and resulting lift are obtained from numerical resolution of the Navier–Stokes equations. The controller is modelled as an artificial neural network and trained to minimise the lift fluctuation using deep reinforcement learning (DRL). The paper shows that DRL-trained controllers are able to mitigate medium- and high-intensity vortex gusts by more than 80 % compared to the uncontrolled scenario. It then presents a comparative analysis of the controlled and uncontrolled lift generation mechanisms using the force partitioning method (FPM). The FPM provides a quantitative assessment of the amount of lift generated by each flow region. For medium-intensity gusts, the main phenomenon is the asymmetry in the airfoil boundary layer induced by the vortex. The control strategy mitigates the gust-induced lift by restoring the flow symmetry around the airfoil. For high-intensity gusts, the boundary layer asymmetry remains, but the gust interaction with the airfoil also triggers flow separation and the formation of a strong leading-edge vortex (LEV). Consequently, the control command balances several aerodynamic phenomena such as boundary layer asymmetry, flow detachment, LEV, and secondary recirculation regions to produce a net quasi-zero lift fluctuation. Thus this work highlights the potential of DRL control, enhanced by advanced post-processing such as FPM, to discover and interpret optimal flow control mechanisms.

For the first time, an analytical solution has been derived for Stokes flow through a conical diffuser under the condition of partial slip. Recurrent relations are obtained that allow determination of the velocity, pressure and stream function for a certain slip length λ. The solution is analysed in the first order of decomposition with respect to a small dimensionless parameter ${\lambda }/{r}$. It is shown that the sliding of the liquid over the surface of the cone leads to a vorticity of the flow. At zero slip length, we obtain the well-known solution to the problem of a diffuser with a no-slip boundary condition corresponding to strictly radial streamlines. To solve that problem, we use an alternative form of the general solution of the linearised, stationary, axisymmetric Navier–Stokes equations for an incompressible fluid in spherical coordinates. A previously published solution to this problem, dating back to the paper by Sampson (1891 Phil. Trans. R. Soc. A, vol. 182, pp. 449–518), is given in terms of a stream function that leads to formulae that are difficult to apply in practice. By contrast, the new general solution is derived in the vector potential representation and is simpler to apply.

The stability of underwater bubbles is important to many natural phenomena and industrial applications. Since stability analyses are complex and influenced by numerous factors, they are often performed on a case-specific basis, with most being qualitative. In this work, we propose a unified and quantitative criterion for evaluating bubble stability by analysing its free energy. This criterion is broadly applicable across various bubble sizes (from nanometres to macroscale) and contact conditions (suspended, attached and trapped bubbles) on surfaces with diverse chemical (hydrophilic and hydrophobic) and morphological (flat and structured solid surfaces) features. This criterion not only applies to the classical stable bubble mode, which depends on contact line pinning at the tips of surface structures, but also predicts a new mode where the synergy between the geometry and wettability of the sidewalls maintains the bubble’s stable state. The contact line can spontaneously adjust its position on the solid surface to maintain pressure balance, which enhances bubble adaptability to environmental changes. A geometric standard for solid surfaces supporting this new stable state is raised, following which we realise the optimisation of solid surface geometries to control the stability of gas bubbles. This work not only provides a universal framework for understanding underwater bubble stability, but also opens avenues for applications.

The actuator line model (ALM) is an approach commonly used to represent lifting and dragging devices like wings and blades in large-eddy simulations (LES). The crux of the ALM is the projection of the actuator point forces onto the LES grid by means of a Gaussian regularisation kernel. The minimum width of the kernel is constrained by the grid size; however, for most practical applications like LES of wind turbines, this value is an order of magnitude larger than the optimal value that maximises accuracy. This discrepancy motivated the development of corrections for the actuator line, which, however, neglect the effect of unsteady spanwise shed vorticity. In this work we develop a model for the impact of spanwise shed vorticity on the unsteady loading of an aerofoil modelled as a Gaussian body force distribution, where the model is applicable within the regime of unsteady attached flow. The model solution is derived both in the time and frequency domain and features an explicit dependence on the Gaussian kernel width. We verify the model with ALM-LES for both pitch steps and periodic pitching. The model solution is compared withTheodorsen theory and validated with both computational fluid dynamics using body fitted grids and experiment. It is concluded that the optimal kernel width for unsteady aerodynamics is approximately $40\,\%$ of the chord. The ALM is able to predict the magnitude of the unsteady loading up to a reduced frequency of $k\approx 0.2$.

We report experiments in a long tank showing that transverse Benjamin–Feir instability of Stokes waves can lead to a significant energy transfer into oscillations across the tank. We observe frequency downshift in the long-term evolution of Stokes waves essentially when significant energy is transferred to narrow-banded transverse modes. Experiments with Stokes waves are often carried out with wavelengths that are not long compared with the width of the tank, permitting transverse instabilities to be excited. With insufficient resolution of measurements across the tank, transfer of energy to transverse modes can be misinterpreted as dissipation. Our experiments suggest that the frequency downshift depends as much on energy-preserving transverse modulations of type I as it does on damping or wave breaking. Broad-banded unstable modulations of type II do not imply downshift.

Magnets have been utilised widely for their ability to induce rapid contact – such as snapping between magnets and ferromagnetic materials. Yet, how such interactions proceed under immersion in a viscous fluid remains poorly understood. Here, we study this problem using the classical configuration of a smooth solid sphere approaching a plane in a quiescent fluid. Induced magnetic attraction, a spatially varying force analogous to short-range dispersion forces, offers a plausible route to overcome the constraint of a diverging hydrodynamic drag, which is well understood using the framework of classical lubrication theory. Instinctively, one might expect it to enable finite-time contact. However, our experiments reveal a counterintuitive result: while magnetic forces accelerate the sphere towards the surface, reducing the approach time by two orders of magnitude compared with gravity, they ultimately fail to effectuate contact in finite time, as induced magnetic interactions are unable to mitigate lubrication drag, which is singular at the thin gap limit, and transitions to an exponential descent characteristic of constant forcing. We support these findings with a simple theoretical model that accurately predicts the magnetic force law from purely kinematic observations. Finally, we outline the conditions under which spatially varying forces can enable true finite-time contact and discuss future experimental directions.

This work investigates the formation mechanism of the turbulent secondary vortex street (SVS) which appears in the far wake of bluff bodies, when the (primary) Kármán vortex street is absent. The turbulent wakes of four types of highly porous bluff bodies (plates/meshes) are characterised via time-resolved particle image velocimetry and large eddy simulations. The effect of ambient turbulence and initial conditions on SVS development is also examined, by installing a turbulence grid upstream of the bodies, and by varying the homogeneity of the bluff body porosity. Our results indicate that the SVS is a far-wake evolution of near-wake shear-layer vortices which, in the absence of the vortex shedding instability, continually grow and are finally arranged into alternating vortices. Free-stream turbulence and body inhomogeneity are found to significantly influence SVS development by amplifying mixing and attenuating the shear-layer instabilities of the near wake, which in turn lead to the formation of weaker and less coherent SVS structures further downstream.

We investigate the evolution of an external particle jet in a dense particle bed subjected to a radially divergent air-blast. Both random and single-mode perturbations are considered. By analysing the particle dynamics, we show that the Rayleigh–Taylor instability (RTI), the Richtmyer–Meshkov instability (RMI) and large particle inertia contribute to the formation of the external jet. The external particle jet exhibits a spike-like structure at its top and a bubble-like structure near its bottom. As the expanding particle bed lowers the internal gas pressure, particles near the bubble experience strong inward coupling forces and undergo RTI with variable acceleration. Meanwhile, particles in the spike experience weak gas–particle coupling and collision forces due to large particle inertia and low particle volume fraction, respectively. Consequently, the particles in the spike retain a nearly constant velocity, in contrast to the accelerating spikes observed in cylindrical RTI. To investigate the contributions of RMI to the particle jet growth, we track the trough-near particles in the single-mode perturbation case. It is revealed that the trough-near particles accelerate under the perturbation-induced pressure gradient, overtaking the crest-near particles and inducing phase inversion, thereby resulting in an increase in jet length. We establish a linear-growth model for the jet length increment, similar to the planar Richtmyer–Meshkov impulsive model. Combined with the jet-length-increment model, we propose an external-particle-jet-length model that is consistent with both numerical and experimental results for diverse initial gas pocket central pressures and particle bed thicknesses.

Asymptotic flow states with limiting drag modification are explored via direct numerical simulations in a moderate-curvature viscoelastic Taylor–Couette flow of the FENE-P fluid. We show that asymptotic drag modification (ADM) states are achieved at different solvent-to-total viscosity ratios ($\beta$) by gradually increasing the Weissenberg number from 10 to 150. As $\beta$ decreases from 0.99 to 0.90, for the first time, a continuous transition pathway is realised from the maximum drag reduction to the maximum drag enhancement, revealing a complete phase diagram of the ADM states. This transition originates from the competition between Reynolds stress reduction and polymer stress development, namely, a mechanistic change in angular momentum transport. Reduced $\beta$ has been found to effectively enhance elastic instability, suppressing large-scale Taylor vortices while promoting the formation of small-scale elastic Görtler vortices. The enhancement and in turn dominance of small-scale structures result in stronger incoherent transport, facilitating efficient mixing and substantial polymer stress development that ultimately drives the AMD state transition. Further analysis of the scale-decomposed transport equation of turbulent kinetic energy reveals an inverse energy cascade in the gap centre, which is attributed to the polymer-induced energy redistribution: polymers extract more energy from large scales than they can dissipate, with the excess energy redirected to smaller scales. However, the energy accumulating at smaller scales cannot be dissipated immediately and is consequently transferred back to larger scales via nonlinear interactions, thereby unravelling a novel polymer-mediated cycle for the reverse energy cascade. Overall, this study unravels the challenging puzzle of the existence of distinct dynamically connected ADM states and paves the way for coordinated experimental, simulation and theoretical studies of transition pathways to desired ADM states.

The guided-jet waves (GJWs) that may be trapped into a jet are investigated by simulating the propagation of the waves generated by an acoustic source on the axis of a jet at a Mach number of 0.95. The flow is modelled as a cylindrical shear layer to avoid reflections in the axial direction. For the source frequencies considered, GJWs belonging to the first two radial GJW axisymmetric modes are observed. They propagate in the upstream or downstream directions, and are entirely or partially contained in the flow, depending on the frequency. Their amplitudes are quantified. In the frequency–wavenumber space, they lie along the GJW dispersion curves predicted using linear-stability analysis. At specific spatial locations, they vary strongly and sharply with the frequency, exhibiting tonal-like peaks near the frequencies of the stationary points in the dispersion curves where the GJWs are standing waves with zero group velocity. Given the flow configuration, these properties can be attributed to propagation effects not requiring axial resonance between upstream- and downstream-travelling waves. Finally, it can be noted that, upstream of the source, outside the jet, the GJW amplitudes fluctuate in a reverse sawtooth manner with very intense peaks up to 30 dB higher than the levels obtained without flow at 10 jet radii from the source, similarly to the GJW footprints in the near-nozzle spectra of high-subsonic jets.

The growth of wall-mounted ice within channel flow which leads to a constriction is of significant practical relevance, especially in applications relating to aero-icing, large-scale pipe networks and mechanical systems. Whilst earlier works have treated ice constrictions as independent of the oncoming flow, few models explicitly account for the two-way coupling between the thermal and dynamical properties of the fluid and the evolving ice. To this end, the present work seeks to describe the interaction between high-Reynolds-number channel flow and constricting ice boundaries governed by Stefan conditions. Numerical simulations of the model indeed reveal that ice forming on the channel walls grows inwards towards the centreline and subsequently creates almost total constriction. In other parameter regimes, however, there is no ice formation. Using both a numerical and asymptotic approach, we identify regions of parameter space in which ice formation, and subsequently flow constriction, does or does not occur.

The clustering of inertial particles in turbulent flows is ubiquitous in many applications. This phenomenon is attributed to the influence of multiscale vortex structures in turbulent flows on particle motion. In this study, our primary goal is to further investigate the vortex effect on particle motion. We perform analytical and numerical simulations to examine the motion of particles in a counter-rotating vortex pair (CVP) with circulation ratio $\gamma \in (-1,0)$. The small, dilute, heavy inertial particles with a low particle Reynolds number are considered. In particular, the particle Stokes number and density factor satisfy $St\in (0,0.3)$ and $ R\in (0,1)$, respectively. We validate the existence of a particle-attracting ring within the CVP, which provides a simple mechanism for particle trapping. Meanwhile, there exists a critical Stokes number $St_{{cr}}$ limiting the occurrence of particle trapping. We provide a formula to predict the value of $St_{{cr}}$, which depends on both $\gamma$ and $R$. Only when $St\lt St_{{cr}}$ can the attracting ring trap the particle initially located within its basin of attraction and eventually lead to the formation of a particle clustering ring. Particles with a larger $R$ are more likely to be trapped in the CVP. While $St\gt St_{{cr}}$, the dynamics of the particles exhibits finite-time ‘leakage’. The attracting ring in the phase space coincides with the saddle point from which particles escape. Although all particles eventually escape, some may remain trapped in the vortex core region for a duration (represented by residence time). The distribution of residence time exhibits a localised exponential-like feature, indicating transient chaos.

Inspired by small intestine motility, we investigate the flow induced by a propagating pendular wave along the walls of a channel lined with rigid, villi-like microstructures. The villi undergo harmonic axial oscillations with a phase lag relative to their neighbours, generating travelling patterns of intervillous contraction. Using two-dimensional lattice Boltzmann simulations, we resolve the flow within the villi zone and the lumen, sampling small to moderate Womersley numbers. We uncover a mixing boundary layer (MBL) just above the villi, composed of semi-vortical structures that travel with the imposed wave. In the lumen, an axial steady flow emerges, surprisingly oriented opposite to the wave propagation direction, contrary to canonical peristaltic flows. We attribute this flow reversal to the non-reciprocal trajectories of fluid trapped between adjacent villi and derive a geometric scaling law that captures its magnitude in the Stokes regime. The MBL thickness is found to depend solely on the wave kinematics given by intervillous phase lag in the low-inertia limit. Above a critical threshold, oscillatory inertia induces dynamic confinement, limiting the radial extent of the MBL and leading to non-monotonic behaviour of the axial steady flux. We further develop an effective boundary condition at the villus tips, incorporating both steady and oscillatory components across relevant spatial scales. This framework enables coarse-grained simulations of intestinal flows without resolving individual villi. Our results shed light on the interplay among active microstructure, pendular wave and finite inertia in biological flows, and suggests new avenues for flow control in biomimetic and microfluidic systems.

The electrokinetic and unstable behaviour near strongly polarised surfaces cannot be well captured by the canonical asymptotic theory for induced-charge electro-osmosis, and the intrinsic mechanism remains unclear. Using direct numerical simulations and scaling analysis, this paper reveals that, near the strongly polarised surfaces, the strong electric double layer charging induces a strong local electric field, which drives the cations in the electrical double layer to extend to a finite region and form an extended space-charge (ESC) layer. The ESC triggers flow instability near strongly polarised surfaces, causing a transition of the velocity scaling exponent in the electric field dependence from a 2 to a 4/3 power law. The findings and mechanisms pave the way for designs of energy and biomedical systems.