Equilibrium, travelling-wave and periodic-orbit solutions of the Navier–Stokes equations provide a promising avenue for investigating the structure, dynamics and statistics of transitional flows. Many such invariant solutions have been computed for wall-bounded shear flows, including plane Couette, plane Poiseuille and pipe flow. However, the organisation of invariant solutions is not well understood. In this paper we focus on the role of symmetries in the organisation and computation of invariant solutions of plane Poiseuille flow. We show that enforcing symmetries while computing invariant solutions increases the efficiency of the numerical methods, and that redundancies between search spaces can be eliminated by consideration of equivalence relations between symmetry subgroups. We determine all symmetry subgroups of plane Poiseuille flow in a doubly periodic domain up to translations by half the periodic lengths and classify the subgroups into equivalence classes, each of which represents a physically distinct set of symmetries and an associated set of physically distinct invariant solutions. We calculate fifteen new travelling waves of plane Poiseuille flow in seven distinct symmetry groups and discuss their relevance to the dynamics of transitional turbulence. We present a few examples of subgroups with fractional shifts other than half the periodic lengths and one travelling-wave solution whose symmetry involves shifts by one third of the periodic lengths. We conclude with a discussion and some open questions about the role of symmetry in the behaviour of shear flows.

The study of the shape of droplets on surfaces is an important problem in the physics of fluids and has applications in multiple industries, from agrichemical spraying to microfluidic devices. Motivated by these real-world applications, computational predictions for droplet shapes on complex substrates – rough and chemically heterogeneous surfaces – are desired. Grid-based discretisations in axisymmetric coordinates form the basis of well-established numerical solution methods in this area, but when the problem is not axisymmetric, the shape of the contact line and the distribution of the contact angle around it are unknown. Recently, particle methods, such as pairwise-force smoothed particle hydrodynamics (PF-SPH), have been used to conveniently forego explicit enforcement of the contact angle. The pairwise-force model, however, is far from mature, and there is no consensus in the literature on the choice of pairwise-force profile. We propose a new pair of polynomial force profiles with a simple motivation and validate the PF-SPH model in both static and dynamic tests. We demonstrate its capabilities by computing droplet shapes on a physically structured surface, a surface with a hydrophilic stripe and a virtual wheat leaf with both micro-scale roughness and variable wettability. We anticipate that this model can be extended to dynamic scenarios, such as droplet spreading or impaction, in the future.

The impact of freestream turbulence (FST) on the aerodynamic performance of a flexible finite wing and the produced wingtip vortex was investigated. The wing had a NACA 4412 airfoil profile and the chord-based Reynolds number was $1.4\times 10^{5}$. The experiments were conducted in a closed-loop wind tunnel with four different inflow turbulence intensities ($0.2\,\%$, $3\,\%$, $8\,\%$ and $13\,\%$) generated using an active turbulence grid. Force balance measurements revealed that increasing the scale of the FST increased the maximum lift and delayed stall. Digital image correlation (DIC) measured deflections of the wing’s structure. Spanwise bending was found to be the dominant deformation. While the wing vibrated at its natural frequency in all conditions, FST increased the amplitude of the vibrations. A similar spectral signature was observed in the lift force fluctuations as well. Stereoscopic particle image velocimetry measurements were obtained two chord lengths downstream of the trailing edge simultaneously with DIC. FST decreased the vortex strength, and marginally increased vortex diffusion and size. It also increased the vortex meandering amplitude, while reducing the meandering frequency band. For the cases with a turbulence intensity of $8\,\%$ and $13\,\%$, the frequency of meandering and the wing’s vibration were similar and a phase relation between the two motions was observed. Proper orthogonal decomposition of the vortex (after removing meandering) and the subsequent velocity field reconstruction revealed temporal fluctuations in the vortex strength at the same frequency as the wing’s vibration. This was linked to the lift force fluctuations induced by the wing’s deformations.

We must address active matter in the context of soft boundaries to bridge the gap between our understanding of active matter and the dynamics of biological systems (represented as active matter) under natural conditions. However, the physics of such active drops (matter) in contact with a soft and deformable surface has remained elusive. In this paper, we attempt to fill this gap and develop a theory for soft, active wetting. Our theory, which accounts for the various free energies for passive substrate and active drops as well as the active stresses, provides an equilibrium description of (active) particle orientation inside the drop and an equilibrium shape of the drop–soft-solid system. We obtain an analytical equation relating the activity to the internal pressure of an active drop. The equilibrium calculation further yields an ordered state of the polarisation field inside the drop. As compared to the non-active drops, the active drops with extensile activity press more into the soft surface, while the active drops with contractile activity either rise out of the soft surface (for smaller magnitude of negative activity) or make the soft surface bulge (for larger magnitude of negative activity). Finally, the three-phase contact line undergoes a rotation that depends on the strength of activity. These findings shed light on the manner in which the active stresses interact with surface tension and elasticity at the fundamental level.

The interaction between the flow in a channel with multiple obstructions on the bottom and an elastic ice sheet covering the liquid is studied for the case of steady flow. The mathematical model employs velocity potential theory and fully accounts for the nonlinear boundary conditions at the ice/liquid interface and on the channel bottom. The integral hodograph method is used to derive the complex velocity potential of the flow, explicitly containing the velocity magnitude at the interface. This allows the boundary-value problem to be reduced to a system of nonlinear equations for the unknown velocity magnitude at the ice/liquid interface, which is solved using the collocation method. Case studies are carried out for a widened rectangular obstruction, whose width exceeds the wavelength of the interface, and for arrays of triangular ripples forming the undulating bottom shape. The influence of the bottom shape on the interface is investigated for three flow regimes: the subcritical regime, $F \lt F_{{cr}}$, for which the depth-based Froude number is less than the critical Froude number, and the interface perturbation decays upstream and downstream of the obstruction; the ice-supercritical and channel-subcritical regime, $F_{cr} \lt F \lt 1$, for which two waves of different wavelengths extend upstream and downstream to infinity; and the channel-supercritical regime, $F \gt 1$, for which the hydroelastic wave extends downstream to infinity. The results revealed a trapped interface wave above the rectangular obstruction and the ripple patch. The resonance behaviour of the interface over the undulating bottom occurs when the period of ripples approaches the wavelength of the ice/liquid interface.

Magnetohydrodynamic turbulence with Hall effects is ubiquitous in heliophysics and plasma physics. Direct numerical simulations reveal that, when the forcing scale is comparable to the ion inertial scale, the Hall effects induce remarkable cross-helicity. It then suppresses the cascade efficiency, leading to the accumulation of large-scale magnetic energy and helicity. The process is accompanied by the disruption of current sheets through the entrainment by vortex tubes or the excitation of whistler waves. Using the solar wind data from the Parker Solar Probe, the numerical findings are separately confirmed. These findings provide new insights into the emergence of large-scale solar wind turbulence driven by helical fields and Hall effects.

Thixotropic fluids with a non-monotonic flow curve display viscosity bifurcations at certain stresses. It has been proposed that these transitions can introduce interfaces (or shear bands) into thin films that can destabilize inertialess flows over inclined planes. This proposition is confirmed in the present paper by formulating a thin-film model, then using this model to construct sheet-like base flows and test their linear stability. It is also found that viscosity bifurcations, and the associated interfaces, are not necessary for instability, but that the time-dependent relaxation of the microstructure responsible for thixotropy within the bulk of the film can promote instability instead. Computations with the thin-film model demonstrate that instabilities saturate supercritically into steadily propagating nonlinear waves that travel faster than the mean flow.

We consider the vortex–wedge interaction problem, taking as a departure point Howe’s model of a point vortex interacting with a semi-infinite half-plane, where the vortex path is influenced by its image and a closed-form analytical solution is obtained for the sound field. We generalise Howe’s model to consider wedges of arbitrary angles and explore the influence of vortex circulation, distance from the edge and the wedge half-angle. The effect of wedge angle on sound emission involves a reduced amplitude of the latter as the former is increased. An extension of the model is proposed to account for convection effects by a non-zero ambient flow. We identify a non-dimensional parameter that characterises the vortex kinematics close to the edge and the associated acoustic effect: high and low values of the parameter correspond, respectively, to high- and low-amplitude sound emission of high and low frequency.

We report our finding from direct numerical simulations that polygonal cell structures are formed by inertial particles in turbulent Rayleigh–Bénard convection in a large aspect ratio channel at Rayleigh numbers of $10^6, 10^7$ and $10^8$, and Prandtl number of 0.7. The settling of small particles modified the flow structures only through momentum interactions. From the simulations performed for various sizes and mass loadings of particles, we discovered that for small- and intermediate-sized particles, cell structures such as square, pentagonal or hexagonal cells were observed, whereas a roll structure was formed by large particles. As the mass loading increased, the sizes of the cells or rolls decreased for all particle sizes. The Nusselt number increased with the mass loading of intermediate and large particles, whereas it decreased with the mass loading of small particles compared with the value for particle-free convection. A detailed investigation of the effective feedback forces of the settling particles inside the hot and cold plumes near the walls revealed that the feedback forces break the up–down symmetry between the hot and cold plumes near the surfaces. This enhances the hot plume ascent while not affecting the cold plume, which leads to the preferred formation of cellular structures. The energy budget analysis provides a detailed interaction between particles and fluid, revealing that the net energy is transferred from the fluid to particles when the particles are small, while settling intermediate and large particles drag the fluid so strongly that energy is transferred from particles to fluid.

Riparian vegetation along riverbanks and seagrass along coastlines interact with water currents, significantly altering their flow. To characterise the turbulent fluid motion along the streamwise-edge of a region covered by submerged vegetation (canopy), we perform direct numerical simulations of a half-channel partially obstructed by flexible stems, vertically clamped to the bottom wall. An intense streamwise vortex forms along the canopy edge, drawing high-momentum fluid into the side of the canopy and ejecting low-momentum fluid from the canopy tip, in an upwelling close to the canopy edge. This mechanism has a profound impact on the mean flow and on the exchange of momentum between the fluid and the structure, which we thoroughly characterise. The signature of the canopy-edge vortex is also found in the dynamical response of the stems, assessed for two different values of their flexibility. Varying the flexibility of the stems, we observe how different turbulent structures over the canopy are affected, while the canopy-edge vortex does not exhibit major modifications. Our results provide a better understanding of the flow in fluvial and coastal environments, informing engineering solutions aimed at containing the water flow and protecting banks and coasts from erosion.

This study suggests that partial changes in adverse pressure gradient (APG) turbulent boundary layers (TBLs) relative to zero pressure gradient (ZPG) conditions can be obtained quantitatively by the wall-normal integral, while clarifying the partial influence of non-equilibrium effects. Specifically, the term $u_{\tau }^{2}/ ( {U_{e}V_{e}} )$, which is found to describe the degree of scale separation under non-equilibrium conditions, is decomposed into three terms. Here, $u_{\tau }$ is the frictional velocity, $U_{e}$ is the streamwise velocity at the boundary layer edge, and $V_{e}$ is the normal velocity at the boundary layer edge. This equation includes a ZPG term, a pressure gradient term and a streamwise variation term, indicating that the pressure gradient promotes scale separation. The equation can be applied to ZPG TBLs and equilibrium APG TBLs by separately ignoring the pressure gradient term and the streamwise variation term. By using this equation to simplify the integral of the inertia term of the mean momentum equation, an expression for the Reynolds shear stress in the outer region can be obtained, which indicates how APG affects the Reynolds shear stress through the mean velocity. The above quantitative results support further study of non-equilibrium APG TBLs.

The Lamb–Oseen vortex is a model for practical vortical flows with a finite vortex core. Vortices with a Lamb–Oseen vortex velocity profile are stable according to the Rayleigh criterion in an infinite domain. Practical situations introduce boundary conditions over finite domains. Direct numerical simulations are performed on the evolution of perturbations to a viscous Lamb–Oseen vortex with uniform inlet axial velocity in a pipe of finite length. Linear stability boundaries are determined in the $(\textit{Re},\omega )$ plane. For a given swirl ratio $\omega$, the flow is found to become linearly unstable when the Reynolds number $\textit{Re}$ is above a critical value. The complete evolution history of the flow is followed until it reaches its final state. For small swirl ratios, the axisymmetric mode is linearly unstable and evolves to a final steady axisymmetric but non-columnar accelerated flow state after nonlinear saturation. For large swirl ratios, the spiral mode is linearly unstable. The spiral mode is found to force growth of an axisymmetric component due to nonlinear interaction. The flow evolves to a final unsteady spiral vortex breakdown state after it undergoes nonlinear saturation. The energy transfer between the mean flow and perturbations is studied by the Reynolds–Orr equation. The pressure work at the exit of the finite pipe is a major source of energy production. Finite-domain boundary conditions also modify the perturbation mode shapes, which can render the vortex core from absorbing energy to producing energy, and thus lead to instabilities. As the pipe length increases, the stability behaviour of the flow is found to approach that predicted by the classical Rayleigh criterion.

This paper provides direct experimental evidence for the coexistence of both a laminar separation bubble and a secondary vortex on the advancing side of a rotating sphere when subjected to the inverse Magnus effect. Detailed experiments were conducted in a wind tunnel on two spheres of varying surface roughness to investigate both ordinary and inverse Magnus effects. Experiments took place for $0.5\times 10^{5}\leqslant {\textit{Re}}\leqslant 3\times 10^{5}$ and rotation rates $0\leqslant \alpha \leqslant 0.45$, where the spheres were rotated via a shaft that was oriented perpendicularly to the free stream flow. Static pressure measurements were made on the non-shaft hemisphere using a spline of taps spanning from the equator to the pole. The ordinary Magnus effect was generally observed at the lowest ${\textit{Re}}$ tested, with a transition to the inverse Magnus effect occurring as ${\textit{Re}}$ increased. Time-averaged pressure coefficient distributions across the equatorial plane were obtained for the smooth and rough spheres. Cross-flow particle image velocimetry was used to visualise the downstream wake velocity field. A pair of counter-rotating wing-tip-like vortices were detected when the sphere experienced the ordinary Magnus effect, generated by flow leakage from the advancing to the retreating side. When the sphere experienced the inverse Magnus effect, the polarity of the counter-rotating vortex pair reversed. This is the first experimental observation of the vortex polarity reversal associated with the inverse Magnus effect in the wake of a rotating sphere. The results provide qualitative visualisation of the complex fluid dynamics and inform future applications of the Magnus effect.

Within the context of machine learning-based closure mappings for Reynolds-averaged Navier Stokes turbulence modelling, physical realisability is often enforced using ad hoc postprocessing of the predicted anisotropy tensor. In this study, we address the realisability issue via a new physics-based loss function that penalises non-realisable results during training, thereby embedding a preference for realisable predictions into the model. Additionally, we propose a new framework for data-driven turbulence modelling which retains the stability and conditioning of optimal eddy viscosity-based approaches while embedding equivariance. Several modifications to the tensor basis neural network to enhance training and testing stability are proposed. We demonstrate the conditioning, stability and generalisation of the new framework and model architecture on three flows: flow over a flat plate, flow over periodic hills and flow through a square duct. The realisability-informed loss function is demonstrated to significantly increase the number of realisable predictions made by the model when generalising to a new flow configuration. Altogether, the proposed framework enables the training of stable and equivariant anisotropy mappings, with more physically realisable predictions on new data. We make our code available for use and modification by others. Moreover, as part of this study, we explore the applicability of Kolmogorov–Arnold networks to turbulence modelling, assessing its potential to address nonlinear mappings in the anisotropy tensor predictions and demonstrating promising results for the flat plate case.

Steady flow at low Reynolds (Re) number through a planar channel with converging or diverging width is investigated in this study. Along the primary direction of flow, the small dimension of the channel cross-section remains constant while the sidewalls bounding the larger dimension are oriented at a constant angle. Due in part to ease of manufacturing, parallel-plate geometries such as this have found widespread use in microfluidic devices for mixing, heat exchange, flow control and flow patterning at small length scales. Previous analytical solutions for flows of this nature have required the converging or diverging aspect of the channel to be gradual. In this work, we derive a matched asymptotic solution, validated against numerical modelling results, that is valid for any sidewall angle, without requiring the channel width to vary gradually. To accomplish this, a cylindrical coordinate system defined by the angle of convergence between the channel sidewalls is considered. From the mathematical form of the composite expansion, a delineation between two secondary flow components emerges naturally. The results of this work show how one of these two components, originating from viscous shear near the channel sidewalls, corresponds to convective mixing, whereas the other component impresses the sidewall geometry on streamlines in the outer flow.

The early stage of a gravity-driven flow resulting from the sudden removal of a floating body is investigated. Initially, the fluid is at rest, with a rigid, symmetric wedge floating on its surface. The study focuses on the initial evolution of the wedge-shaped depression formed on the water’s free surface. The fluid has finite depth, and the resulting flow is assumed to be governed by potential theory. The initial flow is described by a linear boundary-value problem, which is solved using conformal mapping and the theory of complex analytic functions. The behaviour of the flow velocity near the corner points of the fluid domain is analysed in detail. It is shown that the linear theory predicts a power-law singularity in the flow velocity at the vertex of the wedge-shaped depression, with the exponent depending on the wedge angle. As the cavity extends toward the bottom, the flow singularity at the vertex becomes stronger. The local flow near the vertex is shown to be self-similar at leading order in the short-time limit. At the other two corner points – where the initial free surface intersects the surface of the wedge – the linear theory predicts continuous velocities with singular velocity gradients. Theoretical predictions are compared with numerical results obtained using OpenFOAM. Good agreement is observed at short times, except in small vicinities of the corner points, where inner solutions are required. In practical applications, understanding the short-time behaviour of the depressions is important for predicting jet formation in regions of high surface curvature.

The interface shape near a moving contact line is described by the Cox–Voinov theory, which contains a constant term that is not trivially obtained. In this work, an approximate expression of this term in explicit form is derived under the condition of a Navier slip. Introducing the approximation of a local slippery wedge flow, we first propose a novel form of the generalised lubrication equation. A matched asymptotic analysis of this equation yields the Cox–Voinov relation with the constant term expressed in elementary functions. For various viscosity ratios and contact angles, the theoretical predictions are rigorously validated against full numerical solutions of the Stokes equations and available asymptotic results.

This work combines Navier–Stokes–Korteweg dynamics and rare event techniques to investigate the transition pathways and times of vapour bubble nucleation in metastable liquids under homogeneous and heterogeneous conditions. The nucleation pathways deviate from classical theory, showing that bubble volume alone is an inadequate reaction coordinate. The nucleation mechanism is driven by long-wavelength fluctuations with densities slightly different from the metastable liquid. We propose a new strategy to evaluate the typical nucleation times by inferring the diffusion coefficients from hydrodynamics. The methodology is validated against state-of-the-art nucleation theories in homogeneous conditions, revealing non-trivial, significant effects of surface wettability on heterogeneous nucleation. Notably, homogeneous nucleation is detected at moderate hydrophilic wettabilities despite the presence of a wall, an effect not captured by classical theories but consistent with atomistic simulations. Hydrophobic surfaces, instead, anticipate the spinodal. The proposed approach is fairly general and, despite the paper discussing results for a prototypical fluid, it can be easily extended, also in complex geometries, to any real fluid provided the equation of state is available, paving the way to model complex nucleation problems in real systems.

Interactions of turbulent boundary layers with a compliant surface are investigated experimentally at Reτ = 3300–8900. Integrating tomographic particle tracking with Mach–Zehnder interferometry enables simultaneous mapping of the compliant wall deformation and the three-dimensional velocity and pressure fields. Our initial study (J. Fluid. Mech. vol. 980, R2) shows that the flow–deformation correlations decrease with increasing Reτ, despite an order of magnitude increase in deformation amplitude. To elucidate the mechanisms involved, the same velocity, pressure and kinetic energy fields are decomposed to ‘wave-coherent’ and ‘stochastic’ parts using a Hilbert projection method. The phase dependent coherent variables, especially the pressure, are highly correlated with the wave, but decrease with increasing Reτ. While the coherent energy is 6 %–10 % of the stochastic level, the pressure root mean square is comparable near the wall. The energy flux between the coherent and stochastic parts and the pressure diffusion reverse sign at the critical layer. To explain the Reτ dependence, the characteristic deformation wavelength (three times the thickness) is compared with the scales of the energy-containing eddies in the boundary layer represented by the k−1 range in the energy spectrum. When the deformation wavelength is matched with the kxEuu peak at the present lowest Reτ, the flow–deformation correlations and coherent pressure become strong, even for submicron deformations. In this case, the flow and wall motion become phase locked, suggesting resonant behaviours. As Reτ increases, the wall wavelengths and spectral range of attached eddies are no longer matched, resulting in reduced correlations and lower coherent energy and pressure, despite larger deformation.

We consider the efficiency of turbulence, a dimensionless parameter that characterises the fraction of the input energy stored in a turbulent flow field. We first show that the inverse of the efficiency provides an upper bound for the dimensionless energy injection in a turbulent flow. We analyse the efficiency of turbulence for different flows using numerical and experimental data. Our analysis suggests that efficiency is bounded from above, and, in some cases, saturates following a power law reminiscent of phase transitions and bifurcations. We show that for the von Kármán flow the efficiency saturation is insensitive to the details of the forcing impellers. In the case of Rayleigh–Bénard convection, we show that within the Grossmann and Lohse model, the efficiency saturates in the inviscid limit, while the dimensionless kinetic energy injection/dissipation goes to zero. In the case of pipe flow, we show that saturation of the efficiency cannot be excluded, but would be incompatible with the Prandtl law of the drag friction coefficient. Furthermore, if the power-law behaviour holds for the efficiency saturation, it can explain the kinetic energy and the energy dissipation defect laws proposed for shear flows. Efficiency saturation is an interesting empirical property of turbulence that may help in evaluating the ‘closeness’ of experimental and numerical data to the true turbulent regime, wherein the kinetic energy saturates to its inviscid limit.