Turbulent pipe flow is of substantial importance in practical applications, and it remains challenging to depict the characteristic complex multiscale dynamics by a unified theoretical framework, hindered by its inherent intermittency. Inspired by a recent study of velocity circulation in turbulent channel flows from Duan, Chen & Sreenivasan (2025 J. Fluid Mech., vol. 1009, p. R4), in this study, we investigate the statistical characteristics of velocity circulation (or equally the area integral of wall-normal vorticity) over rectangular loops in concentric cylindrical shells, parallel to the pipe wall. The statistics are implemented using direct numerical simulation data at friction Reynolds numbers of $ \textit{Re}_\tau =1057$ and $2000$. Close to the pipe wall, the circulation in the inertial range resides on space-filling unifractal sets, with the Hölder exponent smaller than Kolmogorov’s $4/3$. Away from the pipe wall, the circulation displays bifractal characteristics and the Hölder exponents for high moment orders are very close to those reported in channel flows and homogeneous isotropic turbulence. The circulation statistics are only dependent on the area enclosed by the loops, and are invariant to the loop aspect ratio, once both edge lengths of the loops are in the inertial range.

Traditional Reynolds-averaged Navier–Stokes (RANS) closures, based on the Boussinesq eddy-viscosity hypothesis and calibrated on canonical flows, often yield inaccurate predictions of both mean flow and turbulence statistics. Here, we consider flow past a circular cylinder over a range of Reynolds numbers ($3900$–$100\,000$) and Mach numbers ($0$–$0.3$), encompassing incompressible and weakly compressible regimes, with the goal of improving predictions of mean velocity and Reynolds forces. To this end, we assemble a cross-validated dataset comprising hydrodynamic particle image velocimetry (PIV) in a towing tank, aerodynamic PIV in a wind tunnel and high-fidelity spectral element direct numerical simulation and large eddy simulation. Analysis of these data reveals a universal distribution of Reynolds stresses across the parameter space, which provides the foundation for a data-driven closure. We employ physics-informed neural networks (PINNs), trained with the unclosed RANS equations, to infer the velocity field and Reynolds-stress forcing from boundary information alone. The resulting closure, embedded in a forward PINN solver and the numerical solver OpenFOAM, significantly improves RANS predictions of both mean flow and turbulence statistics relative to conventional models.

Electrohydrodynamic (EHD) instabilities at polymer–porous interfaces play a pivotal role in determining interfacial morphology, wettability and pattern formation, with implications for energy storage, diagnostics and flexible electronics. This study presents a comprehensive general linear stability analysis to examine electric-field-induced instabilities at a confined interface between a viscoelastic polymer gel and a saturated porous medium. By coupling Maxwell stresses with a modified Darcy–Brinkman–Kelvin–Voigt framework, the model captures how porous medium-moderated EHD instabilities influence both the onset and dominant instability modes. Key parameters – including the electric Rayleigh number, Darcy number, dielectric contrast and geometric filling ratio – govern the spatio-temporal features of emerging patterns. The analysis reveals a sigmoidal dependence of characteristic length and time scales on permeability, i.e. Darcy number, establishing three regimes: impermeable, transitional and highly permeable, with a shift toward shorter wavelengths. The length and time scale transitions, triggered by the solid-saturated porous medium, are further moderated by the dielectric contrast – instabilities are suppressed when the contrast is low and amplified when it is high, enabling sub-micron patterning. Geometric confinement, i.e. increasing filling ratio, further intensifies pattern length scales, suggesting the feasibility of fabricating complex ultra-fine nanoscale encapsulated porous patterns. The elasticity of the viscoelastic layer imposes a threshold for instability onset and is critical for identifying wettability transitions at the interface. This framework offers predictive insight into tuning instability modes through permeability–viscoelasticity–electrostatics interplay, laying the foundation for wettability-controlled interfaces and self-organised interfacial patterns in next-generation EHD-driven systems.

This paper investigates asymmetric shock reflection in a supersonic overexpanded jet. A theoretical model that can predict the size and shape of flow field structures such as shocks, expansion waves and sliplines is established. For symmetric Mach reflection, the current model exhibits better agreement with numerical simulation results. For asymmetric Mach reflection, the current model also shows good agreement with numerical simulation results in predicting Mach stem height. The research indicates that the Mach stem height decreases with increasing nozzle pressure ratio and nozzle exit length difference, and decreases as the nozzle exit Mach number decreases. In addition, the critical geometric condition for complete misalignment of the upper and lower slipline interference segments (i.e. when the interference between one side of the expansion wave and the slipline ends, the other side of the expansion wave has not yet begun to interfere with the slipline) is given, which increases approximately linearly with nozzle pressure ratio and decreases as the nozzle exit Mach number increases. This study provides important theoretical support for the engineering application of asymmetric shock reflection in supersonic overexpanded jet.

Many environmental, energy and industrial processes involve the flow of viscoelastic polymer solutions in three-dimensional (3-D) porous media where fluid is confined to navigate through complex pore space geometries. As polymers are transported through the tortuous pore space, elastic stresses accumulate, leading to the onset of unsteady, time-dependent flow fluctuations above a threshold flow rate. How does pore space geometry influence the development and features of this elastic instability? Here, we address this question by directly imaging polymer solution flow in microfabricated 3-D ordered porous media with precisely controlled geometries consisting of simple-cubic (SC) or body-centred cuboid (BC) arrays of spherical grains. In both cases, we find that the flow instability is generated at stagnation points arising at the contacts between grains rather than at the polar upstream/downstream grain surfaces, as is the case for flow around a single grain. The characteristics of the flow instability are strongly dependent on the unit cell geometry: in SC packings, the instability manifests through the formation of time-dependent, fluctuating 3-D eddies; whereas in BC packings, it manifests as continual fluctuating ‘wobbles’ and crossing in the flow pathlines. Despite this difference, we find that characteristics of the transition from steady to unsteady flow with increasing flow rate have commonalities across geometries. Moreover, for both packing geometries, our data indicate that extensional flow-induced polymeric stresses generated by contact-associated stagnation points are the primary contributor to the macroscopic resistance to flow across the entire medium. Altogether, our work highlights the pivotal role of inter-grain contacts – which are typically idealised as discrete points and therefore overlooked, but are inherent in most natural and engineered media – in shaping elastic instabilities in porous media.

We investigate mixing dynamics in porous media at finite times, using pore-scale lattice-Boltzmann simulations combined with Lagrangian particle tracking. We compute fluid deformation in randomly packed beds based on the moving Protean frame approach introduced by Lester et al. (2018 J. Fluid Mech. 855, 770–803). From the extracted Lagrangian kinematics, we construct a mixing model based on lamellar aggregation that well predicts the Eulerian scalar fields obtained from simulations. Our results reveal an early-time mixing regime dominated by shear-driven fluid deformation, where solute mixing arises from the random overlap of diffusive concentration elements. In this regime, mixing proceeds slowly and follows a temporal decay of concentration variance, $\sigma _c^2 \propto \textit{Pe}^{-\alpha /(2\alpha +1)} t^{-1/2}$, where $ \textit{Pe}$ is the Péclet number and $\alpha$ the exponent characterising shear deformation. This dynamic arises when the Péclet number is small relative to the ratio between the exponential-mixing and shear-deformation time scales. This analysis also demonstrates that shear-induced mixing governs the homogenisation of early-stage reactions at the fluid–solid interface in finite-size random packed beds, typically operating at moderate Péclet numbers $ \textit{Pe} =O(10^2)$.

We experimentally study a scallop-like swimmer with reciprocally flapping wings in a nearly frictionless, cohesive granular medium consisting of hydrogel spheres. Significant locomotion is found when the swimmer’s flapping frequency matches the inverse relaxation time of the material. Remarkably, the swimmer moves in the opposite direction compared with its motion in a cohesion-free granular material of hard plastic spheres. At higher or lower frequencies, we observe no motion of the swimmer, apart from a short initial transient phase. X-ray radiograms reveal that the wing motions create low-density zones, which in turn give rise to a hysteresis in drag and propulsion forces. This time-dependent effect, combined with the swimmer’s inertia, accounts for locomotion at intermediate frequencies.

Coarse-grained continuous descriptions for lipid bilayers are typically based on minimising the Helfrich energy. Such models consider the fluid properties of these structures only implicitly and have been shown to nicely reproduce equilibrium properties. Model extensions that also address the dynamics of these structures are surface (Navier–)Stokes–Helfrich models. They explicitly account for membrane viscosity. However, these models also usually treat the lipid bilayer as a homogeneous continuum, neglecting the molecular degrees of freedom of the lipids. Here, we derive refined models that consider in addition a scalar order parameter representing the molecular alignment of the lipids along the surface normal. Starting from hydrodynamic surface liquid crystal models, we obtain a hydrodynamic surface Landau–Helfrich model for asymmetric lipid bilayers and a surface Beris–Edwards model for symmetric lipid bilayers. The fully ordered case for both models leads to the known surface (Navier–)Stokes–Helfrich models. Besides more detailed continuous models for lipid bilayers, we therefore also provide an alternative derivation of surface (Navier–)Stokes–Helfrich models. The impact on the dynamics is demonstrated by numerical simulations.

The inner–outer interaction model (IOIM), first proposed by Marusic et al. (Science, 2010, vol. 329, pp. 193–196), has proven to be an effective turbulence model for canonical and non-canonical wall-bounded flows, where a reference velocity signal from the logarithmic region acts as the input for predicting near-wall velocity fluctuations. Its most recent iteration by Baars et al. (Phys. Rev. Fluids, 2016, vol. 1, p. 054406) further proposes a user-independent scale separation point, refining model parameters. In this study, we compared the long-perceived universal IOIM’s parameters, including the linear transfer kernel, amplitude modulation coefficients and the universal signal for a range of Reynolds and Mach numbers, where mathematical relationships between the parameters are proposed. We observed that while the universal signals exhibit a high degree of similarity, particularly near the wall, the amplitude modulation coefficients and linear transfer kernels display Reynolds and Mach number effects, where varying the reference location also causes them to exhibit significant changes. We have found transformations to collapse amplitude modulation coefficients for incompressible flows and differing reference locations, improving modelling via the IOIM across flow parameters. Despite this, compressibility effects cannot be suitably accounted for currently and remain a future challenge for the IOIM framework.

A recent study by Zhang et al. (2024, J. Fluid Mech., vol. 979, A43) introduced an effective control strategy, namely streamwise-uniform spanwise equally distributed injection/suction slots on the pressure (unstable) wall, to enhance passive scalar transport in spanwise rotating plane Poiseuille flows (RPPFs). In this work, we employ direct numerical simulations to further investigate the scalar transport increase rate ($\textit{STI}$) under different slot configurations. Two distinct configurations are investigated, namely uniform-width slots, where injection and suction slots share identical dimensions, and non-uniform-width slots, where their widths vary independently. The former is to examine the effect of slot width, whereas the latter is devoted to distinguishing the individual roles of injection versus suction. While the slot widths change, the root mean square wall-normal velocity is maintained at a fixed minimal value. For uniform configurations, $ \textit{STI}$ increases monotonically with slot width, nearly doubling as the width grows from $\pi /8$ to $\pi /2$. In contrast, non-uniform configurations exhibit a complex, non-monotonic dependence on slot dimensions. Spectral, quadrant and zonal-conditional quadrant analyses reveal that injection and suction slots play distinct roles in modulating near-wall dynamics. Injection enhances ejection events ($Q2$), promoting local plume detachment, and facilitating the formation of large-scale ascending plume currents. Suction, conversely, strengthens sweep events ($Q4$), suppressing plume detachment while intensifying descending currents. This dual mechanism organises turbulent structures into more stable large-scale structures, thereby improving scalar transport efficiency. A decomposition of $ \textit{STI}$ based on clustering analysis confirms that the enhancement stems primarily from increased occurrence possibilities and improved transport capacities of dominant clusters. These findings establish flow stabilisation through selective slot control as an effective mechanism for enhancing passive scalar transport in RPPFs.

This work investigates experimentally and numerically the dynamics of rigid particles with two orthogonal symmetry planes settling under gravity in a highly viscous fluid at a Reynolds number much smaller than one. Joshi & Govindarajan (2025 Phys. Rev. Lett. 134(1), 014002), showed theoretically that for such shapes, the dynamics are qualitatively different for different signs of the product of two rotational–translational mobility coefficients, evaluated with respect to the particle centre of mass in a symmetric reference frame. However, upon examining a particle’s shape, it is not immediately evident if this product is negative, positive or zero. In this paper, we demonstrate how to estimate these coefficients and the sign of their product from experiments, using special initial orientations, and also numerically, based on the Stokes equations. Especially interesting are the ‘settlers’ – such particles that reorient and approach a stationary stable orientation, and we focus our study on this class of shapes. We show experimentally that cones, crescent moons, arrowheads and open flat rings are the settlers, and we evaluate from the experiments their rotational–translational mobility coefficients. Then, we reconstruct each experimental shape as a rigid conglomerate of many touching beads, and use the precise Hydromultipole code to calculate the mobility coefficients for the conglomerate. The numerical and experimental values are close enough to determine that the particles are the settlers, and to estimate the characteristic reorientation time scales. Our findings apply to non-Brownian micro-objects in water-based solutions – experimentally by the similarity principle and theoretically based on the Stokes equations. The reorientation of sedimenting rigid particles to a stationary stable configuration in a relatively short time might be used for environmental, biological, medical or industrial applications.

Contrary to accepted turbulence folklore, which holds that no mathematical relation exists between the Navier–Stokes equations (NSEs) and the multifractal model (MFM) of Parisi and Frisch, we develop a theory that reconciles the MFM with Leray’s weak solutions of Navier–Stokes analysis. From a combination of Euler invariant scaling and the NSEs set in a three-dimensional box of side $L$, we also derive the Paladin–Vulpiani scale $\eta_{h,pav}$ which is related to the Reynolds number Re by $L\eta _{h,\textit{pa}v}^{-1} = \textit{Re}^{1/(1+h)}$, and which acts as a mediator between the two theories. This is achieved by considering $L^{2m}$-norms of the velocity gradient to find a correspondence between $m$ and the local scaling exponent $h$ in the multifractal model. The parameter $m$ acts as if it were the sliding focus control on a telescope which allows us to zoom in and out on different structures. The range $1 \leqslant m \leqslant \infty$ is equivalent to $-{{ {2}/{3}}} \leqslant h_{\textit{min}} \leqslant {{{1}/{3}}}$, which lies precisely in the region where Bandak et al. (Phys. Rev. E, 2022, vol. 105, p. 065113; Phys. Rev. Lett., 2024, vol. 132, p. 104002) have suggested that thermal noise makes the NSEs inadequate and generates spontaneous stochasticity. The implications of this are discussed.

Flow around a submerged cylinder near a free surface reveals that adjusting the Froude number and gap ratio influences the underwater jet pattern, vortex shedding frequency and free-surface deformation. The jet typically separates near the trough, leading to vorticity concentration and breaking waves that dissipate wave energy. Antarctic orcas collaborate to generate deep depression waves, breaking ice and washing seals from floes. Orcas raise their heads and tap their tails downward when approaching ice, which may benefit strong wave generation. We investigate the wave-generating hydrodynamics using a towing tank and particle image velocimetry. A scaled model with an elliptical body and wedge-shaped tail was tested under Froude number similarity. Experiments covered towing speeds of $0.3- 0.7\,\textrm{ms}^{-1}$, combining different body ($10^\circ$/$0^\circ$/$-10^\circ$) and tail angles ($30^\circ$/$0^\circ$/$-30^\circ$), at chord-based Reynolds numbers of $17\,030- 40\,506$. Four wake regimes are identified: small-scale vortex emergence triggered by capillary waves; extensive wave breaking due to flow separation at the trough; smooth depression wave caused by jet reattachment and downward advection of wake vortices; and large-scale vortex impingement generated by wake vortex perturbations. Under the pitched posture, the jet attaches successively to the solid surface and the trough via the Coandâ effect, suppressing flow separation, creating the most pronounced wave. The strong jet maintained a low-potential-energy state of the wave and led to large ice floes flipping and fracturing through the bending effect, while smaller ice floes were overwashed. This study suggests a novel flow-control strategy for objects near the free surface through jet attachment.

Neural network observers (NNOs) are proposed for online estimation of fluid flows, addressing a key challenge in flow control: obtaining flow states online from a limited set of sparse and noisy sensor data. For this task, we propose a generalisation of the classical Luenberger observer. In the present framework, the estimation loop is composed of subsystems modelled as neural networks (NNs). By combining flow information from selected probes and a neural network surrogate model (NNSM) of the flow system, we train NNOs capable of fusing information to provide the best estimation of the states, that can in turn be fed back to a neural network controller (NNC). The NNO capabilities are demonstrated for three nonlinear dynamical systems. First, a variation of the Kuramoto–Sivashinsky (KS) equation with control inputs is studied, where variables are sparsely probed. We show that the NNO is able to track states even when probes are contaminated with random noise or with sensors at insufficient sample rates to match the control time step. Then, a confined cylinder flow is investigated, where velocity signals along the cylinder wake are estimated by using a small set of wall pressure sensors. In both the KS and cylinder problems, we show that the estimated states can be used to enable closed-loop control, taking advantage of stabilising NNCs. Finally, we present a legacy dataset of a turbulent boundary layer experiment, where convolutional NNs are employed to implement the models required for the estimation loop. We show that, by combining low-resolution noise-corrupted sensor data with an imperfect NNSM, it is possible to produce more accurate and robust estimates. Our approach presents better robustness to noise when compared with direct reconstructions via super-resolution NNs and predictions from graph NNs and Fourier neural operators.