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.

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.

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.

The effect of a horizontal magnetic field on heat transport and flow structures in vertical liquid metal convection (Prandtl number $Pr \approx 0.03$) is investigated experimentally. The experiments are carried out for Rayleigh numbers in the range of $1.48 \times 10^6 \leqslant Ra \leqslant 3.54 \times 10^{7}$ and Chandrasekhar numbers in the range of $2 \times 10^2 \leqslant Q \leqslant 1.86 \times 10^6$, as well as for the non-magnetic case ($Q=0$). Measurements of the heat transport show a rise in the Nusselt number at low and moderate magnetic field strengths up to an optimum value of $Q$, before a further increase in the magnetic field leads to a decrease in the transport properties. By applying simultaneous velocity and temperature measurements, we are able to identify three different oscillatory flow regimes for $10^{-5}\lt Q/Ra \lt 0.5$ and assign them to the respective heat transfer characteristics. In the range $10^{-5}\gt Q/Ra\gt 10^{-3}$, first evidence of a transition to anisotropic flow structures caused by the magnetic field is visible. Two strongly oscillatory regimes are identified, where the energy is either distributed around a dominant frequency ($10^{-3}\gt Q/Ra\gt 10^{-2}$), or strongly concentrated on a single frequency ($10^{-2}\gt Q/Ra\gt 0.5$). The dominating frequency increases with the Rayleigh number according to $Ra^{0.71\pm 0.02}$. This flow structure based regime separation correspond to changes of both the heat transfer through the Nusselt number and mass transfer through the Reynolds number.

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.

Turbulent separating and reattaching flows are known to exhibit low-frequency fluctuations manifested in a large-scale contraction and expansion of the reverse-flow region. Previous experimental investigations have been restricted to planar measurements, while the computational cost to resolve the low-frequency spectrum with high-fidelity simulations currently appears to be unaffordable. In this article, we make use of volumetric measurements to reveal the low-frequency dynamics of a turbulent separation bubble (TSB) formed in the fully turbulent flow past a smooth backward-facing ramp. The volumetric velocity field measurements cover the entire separated flow region over a domain with a spanwise extent of $S=0.6\, {\textrm{m}}$. Spectral proper orthogonal decomposition (SPOD) of the velocity fluctuations reveals low-rank low-frequency behaviour at Strouhal numbers ${\textit{St}}\lt 0.05$, which was also observed in previous planar measurements. However, in contrast with the interpretation of a two-dimensional contraction/expansion motion, the low-frequency dynamics is shown to be inherently three-dimensional, and governed by large elongated structures with a spanwise wavelength of approximately $S/2$. A low-order model constructed with the leading SPOD mode confirms substantial changes of the TSB extent in the centre plane, linking it to the modal pattern that is strongly non-uniform in the spanwise direction. The findings presented in this study promote a more complete understanding of the low-frequency dynamics in turbulent separated flows, thereby enabling novel modelling and control approaches.

We present a mathematical solution for the two-dimensional linear problem involving acoustic-gravity waves interacting with rectangular barriers at the bottom of a channel containing a slightly compressible fluid. Our analysis reveals that, below a certain cutoff frequency, the presence of a barrier inhibits the propagation of acoustic-gravity modes. However, through the coupling with evanescent modes existing in the barrier region, we demonstrate the phenomenon of ‘tunnelling’ where the incident acoustic-gravity wave energy can leak to the other side of the barrier, creating a propagating acoustic-gravity mode of the same frequency. Notably, the amplitude of the tunnelling waves exponentially decays with the width of the barrier, analogous to the behaviour observed in quantum tunnelling phenomena. Moreover, a more general solution for multi-barrier and multi-modes is discussed. It is found that tunnelling energy tends to transform from an incident mode to the lowest neighbouring modes. Resonance due to barrier length results in more efficient energy transfer between modes.

This study explores interfacial waves in a three-layer fluid system, focusing on the coupling effects between the two interfaces. These effects include resonance induced by inertial coupling and damping caused by viscous coupling. A linear theoretical framework is developed to describe the coupled wave motion and evaluate the impact of interfacial coupling under viscous damping. Additionally, a semi-analytical model is introduced to accurately capture resonance frequency shifts and phase differences due to viscosity. The spiral structure of interfacial waves predicted by the models is confirmed experimentally using the background oriented Schlieren (BOS) method. Further, the model is validated by excellent agreement between theoretical predictions and ultrasonic measurements of wave amplitudes and phase differences. Finally, the study examines mechanical coupling and energy transfer between interfaces under external forcing, elucidating the formation of spiral waves. The accurate treatment of viscous boundary conditions by the semi-analytical model also enables its extension to multilayer fluid systems.

Large-scale circulation (LSC) dynamics have been studied in thermal convection driven by heat-releasing particles via the four-way coupled Euler–Lagrange approach. We consider a wide range of Rayleigh–Robert number (${\textit{Rr}}=4.97\times 10^{5} - 4.97 \times 10^{8}$) and density ratio ($\hat {\rho }_r=1- 1000$) that characterize the thermal buoyancy and the particle inertia, respectively. An intriguing flow transition has been found as $\hat {\rho }_r$ continuously increases, involving in sequence three typical LSC regimes, i.e. the bulk-flow-up regime, the marginal regime and the bulk-flow-down (BFD) regime. The comprehensive influence of the LSC regime transition is demonstrated by examining the key flow statistics. As integral flow responses, the heat transfer efficiency and flow intensity change substantially when the LSC regime transition happens, and the thermal boundary layer thicknesses at the top and bottom walls exhibit similar alterations. Significant local accumulation of particles occurs as $\hat {\rho }_r$ increases to a sufficiently high value, resulting in a great modification in the flow dynamics. Specifically, particles aggregate near the sidewalls and heat the local surrounding fluid to generate rising warmer plumes that drive the LSC regime transition. Of interest, well-patterned cellular structures of particles take place near the top wall and obtain notable deviation from the thermal convection cells for the BFD regimes. A mechanical interpretation is proposed and substantiated based on a conceptual vortex–particle model, namely, the centrifugal motion of heat-releasing particles that is confirmed to play a driving role for the LSC regime transition.

A novel particle-resolved direct numerical simulations (PR-DNS) method for non-spherical particles is developed and validated in the open-source MFiX (Multi-phase Flow with Interphase eXchanges) code for simulating the suspension of non-spherical particles and fluidisation. The model is implemented by coupling superquadric Discrete Element Method-Computational Fluid Dynamics (DEM-CFD) with the immersed boundary method. The model was first validated by applying it to analyse fluid dynamic coefficients ($C_{\!D} , C_{\!L} , C_{\!T}$) of superellipsoids and cylinders at different Reynolds numbers, and the PR-DNS results closely matched those of previous methods, demonstrating the reliability of the current PR-DNS approach. Then, the model was applied to the simulation of the fluidisation of spheres and cylinders. The PR-DNS results were compared with both particle-unresolved superquadric DEM-CFD simulation and experimental data. The pressure drop, height distribution and orientation distribution of particles were analysed. The results show that the PR-DNS method provides a reliable method for reproducing fluidisation experimental results of non-spherical particles. In addition, the comparison of the drag correction coefficients predicted by existing models with that obtained from PR-DNS results indicates the need for a new drag model for particle-unresolved simulation of non-spherical particles.

The effects of the external intermittent behaviour on the Kolmogorov constants $C_{k1}$ and $C_2$ in spectral and the physical spaces are investigated using high-resolution direct numerical simulations of a turbulent plane jet. Well-defined $- 5/3$ energy spectrum and $2/3$ structure function can be found in the intermittent flows without large-scale vortex shedding. For different cross-wise positions, the profiles of conditional energy spectra and conditional structure functions exhibit self-similarity at small and intermediate scales when normalised by the conditional Kolmogorov scale of the turbulent region. The conditional Kolmogorov constants are close to those of the fully turbulent flow. The constants $C_{k1}$ and $C_2$ are found to have a power-law dependence on the intermittency factor $\gamma$, that is, $C_{k1}\sim \gamma ^{1/3}$ and $C_{2}\sim \gamma ^{1/3}$, except for the scaling of the structure function in the highly intermittent region with $\gamma =0.25$. In the highly intermittent region, e.g. $\gamma =0.25$, the scaling in the conditional structure function can be considerably influenced by the blocking/sheltering mechanisms of the turbulent/non-turbulent interface (TNTI), leading to slight deviations from self-similarity. We further confirm that the conditional structure function recovers self-similarity after excluding a turbulent region at an average distance of approximately $20$ Kolmogorov length scales from the outer edge of the TNTI, which is comparable to the mean thickness of the TNTI. These findings contribute to the modelling of the edge of a turbulent region.

The inertial migration of hydrogel particles suspended in a Newtonian fluid flowing through a square channel is studied both experimentally and numerically. Experimental results demonstrate significant differences in the focusing positions of the deformable and rigid particles, highlighting the role of particle deformability in inertial migration. At low Reynolds numbers (${Re}$), hydrogel particles migrate towards the centre of the channel cross-section, whereas the rigid spheres exhibit negligible lateral motion. At finite ${Re}$, they focus at four points along the diagonals in the downstream cross-section, in contrast to the rigid particles which focus near the centre of the channel face at similar ${Re}$. Numerical simulations using viscous hyperelastic particles as a model for hydrogel particles reproduced the experimental results for the particle distribution with an appropriate Young’s modulus of the hyperelastic particles. Further numerical simulations over a broader range of ${Re}$ and the capillary number ($Ca$) reveal various focusing patterns of the particles in the channel cross-section. The phase transitions between them are discussed in terms of the inertial lift and the lift due to particle deformation, which would act in the direction towards lower shear. The stability of the channel centre is analysed using an asymptotic expansion approach to the migration force at low ${Re}$ and $Ca$. The theoretical analysis predicts the critical condition for the transition, which is consistent with the direct numerical simulation. These experimental, numerical and theoretical results contribute to a deeper understanding of inertial migration of deformable particles.

Rotor–stator interactions in turbomachines are characterised by a complex interplay of hydrodynamic instabilities, acoustic pressure waves and receptivity mechanisms, as well as the collision of coherent structures with the blade geometry. An unsteady dual analysis of self-excited instabilities and flow interactions, exemplified by a simple model compressor stage under subsonic conditions, is proposed and presented. Using a low-dissipation sliding-plane implementation, instability-resolving nonlinear-adjoint looping simulations provide detailed sensitivity information that allows for the dissection of the full flow into sub-components linked to distinct flow phenomena. This sensitivity information further links observed flow behaviour to its hydrodynamic or acoustic origin, thereby laying the foundation for a cause-and-effect analysis and for flow control.

To study the physics of small-scale properties of homogeneous isotropic turbulence at increasingly high Reynolds numbers, direct numerical simulation results have been obtained for forced isotropic turbulence at Taylor-scale Reynolds number $R_\lambda =2500$ on a $32\,768^3$ three-dimensional periodic domain using a GPU pseudo-spectral code on a 1.1 exaflop GPU supercomputer (Frontier). These simulations employ the multi-resolution independent simulation (MRIS) technique (Yeung & Ravikumar 2020, Phys. Rev. Fluids, vol. 5, 110517) where ensemble averaging is performed over multiple short segments initiated from velocity fields at modest resolution, and subsequently taken to higher resolution in both space and time. Reynolds numbers are increased by reducing the viscosity with the large-scale forcing parameters unchanged. Although MRIS segments at the highest resolution for each Reynolds number last for only a few Kolmogorov time scales, small-scale physics in the dissipation range is well captured – for instance, in the probability density functions and higher moments of the dissipation rate and enstrophy density, which appear to show monotonic trends persisting well beyond the Reynolds number range in prior works in the literature. Attainment of range of length and time scales consistent with classical scaling also reinforces the potential utility of the present high-resolution data for studies of short-time-scale turbulence physics at high Reynolds numbers where full-length simulations spanning many large-eddy time scales are still not accessible. A single snapshot of the $32\,768^3$ data is publicly available for further analyses via the Johns Hopkins Turbulence Database.

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.

Wall-based active and passive flow control for drag reduction in low-Reynolds-number (${\textit{Re}}$) turbulent flows can lead to three typical phenomena: (i) attenuation or (ii) amplification of the near-wall cycle, and (iii) generation of spanwise rollers. The present study conducts direct numerical simulations of a low ${\textit{Re}}$ turbulent channel flow and demonstrates that each flow response can be generated with a wall transpiration at two sets of spatial scales, termed streak and roller scales. The effect of the transpiration is controlled by its relative phase to the background flow, which can be parametrised by the wall pressure. Streak scales (i) attenuate the near-wall cycle if transpiration and wall pressure are approximately in-phase or (ii) amplify it otherwise, and (iii) roller scales energise spanwise rollers when transpiration and wall pressure are out-of-phase. Conditions for establishing these robust phase relations are derived from the analytical solution to the pressure Poisson equation and rely on splitting the pressure into its fast, slow and Stokes component. The importance of each condition depends on the relative magnitude of the pressure components, which is significantly altered by the transpiration. The analogy in flow response suggests that transpiration with the two scale families and their phase relations to the wall pressure represent fundamental building blocks for flows over tailored surfaces including riblets, porous and permeable walls.

A discrete Markov model is proposed to study the interscale dynamics of high Reynolds number wall turbulence. The amplitude modulation of the small turbulent scales due to the interaction with large turbulent scales is investigated for three experimental turbulent boundary layers. Through an appropriate discretisation of the turbulence signals, recently proved universal thermodynamic bounds for discrete-state stochastic systems are shown to apply to continuous-state systems like turbulence, regardless of the distance from the wall and the Reynolds number. Adopting Schnakenberg’s network theory for stochastic processes, we provide evidence for a direct proportionality relation between the mean cycle affinity-based entropy production rate (a stochastic thermodynamic quantity) and a mean entropy production rate associated with the net large-to-small-scale turbulent kinetic energy production. Finally, new insights into the relative arrangement (lag/lead) between large and small scales are provided.

The moulting of birds creates different trailing-edge gaps in their wings, which inspires the handling of damaged wings in micro-air vehicles. The effects of the moult gap on aerodynamic performance are investigated by employing a bird-inspired flapping wing model. The aerodynamic performance is evaluated by numerically solving the Navier–Stokes equations for incompressible flows. Moult-gapped wings with different gap widths and positions are compared with the original intact wing in terms of aerodynamic forces and vortex structures. It is found that the decrease in the average lift is slower than that expected from the classical aerodynamic model. The moult gap results in three-dimensional gap vortices, which interact with leading-edge vortices and tip vortices. The interaction generates a pair of parallelly arranged vortex loops on each wing. The downwash momentum associated with this pair of vortex loops is enhanced by the gap vortices. The gap-vortices-enhanced downwash compensates for the loss in the lifting surface, increasing the aerodynamic force per unit area. A composite actuator disk model is proposed based on the vortex loops. The proposed model accounts for not only the finite-span wing effects but also the vortex compensation effects, while the previous quasi-steady model only accounts for the finite-span wing effects.