We present a versatile framework that employs Physics-Informed Neural Networks (PINNs) to discover the entropic contribution that leads to the constitutive equation for the extra-stress in rheological models of dilute polymer solutions. In this framework the training of the neural network is guided by an evolution equation for the conformation tensor, which is GENERIC-compliant. We compare two training methodologies for the data-driven PINN constitutive models: one trained on data from the analytical solution of the Oldroyd-B (OB) model under steady-state rheometric flows (PINN-rheometric), and another trained on in silico data generated from computational fluid dynamics (CFD) simulations of complex flow around a cylinder that use the OB model (PINN-complex). The capacity of the PINN models to provide good predictions is evaluated by comparison with CFD simulations using the underlying OB model as a reference. Both models are capable of predicting flow behaviour in transient and complex conditions; however, the PINN-complex model, trained on a broader range of mixed-flow data, outperforms the PINN-rheometric model in complex flow scenarios. The geometry agnostic character of our methodology allows us to apply the learned PINN models to flows with topologies different from those used for training.

Viscous fingering is a hydrodynamic instability typically occurring when a less viscous fluid displaces a more viscous one and which deforms the interface between the two fluids into finger-shaped intrusions. For miscible fluids, the fingering pattern is usually followed visually by adding a passive dye into one of the two fluids. The reverse displacement of a less viscous fluid by a more viscous one is classically stable, featuring a planar interface. Here, we show experimentally that in some cases, the dye can actively modify the viscosity of a polymer solution and trigger fingering in the reverse displacement. This dye-induced destabilisation is shown to be due to double-diffusive effects triggering a non-monotonic viscosity profile with a maximum because the dye diffuses faster than the polymer.

Direct numerical simulation (DNS) studies of power-law (PL) fluids are performed for purely viscous-shear-thinning ($n\in [0.5,0.75]$), Newtonian ($n=1$) and purely viscous-shear-thickening ($n=2.0$) fluids, considering two Reynolds numbers ($Re_{\tau }\in [395,590]$), and both smooth and rough surfaces. We carefully designed a numerical experiment to isolate key effects and simplify the complex problem of turbulent flow of non-Newtonian fluids over rough surfaces, enabling the development of a theoretical model to explain the observed phenomena and provide predictions. The DNS results of the present work were validated against literature data for smooth and rough Newtonian turbulent flows, as well as smooth shear-thinning cases. A new analytical expression for the mean velocity profile – extending the classical Blasius $1/7$ profile to power-law fluids – was proposed and validated. In contrast to common belief, the decrease in $n$ leads to smaller Kolmogorov length scales and the formation of larger structures, requiring finer grids and longer computational domains for accurate simulations. Our results confirm that purely viscous shear-thinning fluids exhibit drag reduction, while shear-thickening fluids display an opposite trend. Interestingly, we found that viscous-thinning turbulence shares similarities with Newtonian transitional flows, resembling the behaviour of shear-thinning, extensional-thickening viscoelastic fluids. This observation suggests that the extensional and elastic effects in turbulent flows within constant cross-section geometries may not be significant. However, the shear-thickening case exhibits characteristics similar to high-Reynolds-number Newtonian turbulence, suggesting that phenomena observed in such flows could be studied at significantly lower Reynolds numbers, reducing computational costs. In the analysis of rough channels, we found that the recirculation bubble between two roughness elements is mildly influenced by the thinning nature of the fluid. Moreover, we observed that shear-thinning alters the flow in the fully rough regime, where the friction factor typically reaches a plateau. Our results indicate the possibility that, at sufficiently high Reynolds numbers, this plateau may not exist for shear-thinning fluids. Finally, we provide detailed turbulence statistics for different rheologies, allowing, for the first time, an in-depth study of the effects of rheology on turbulent flow over rough surfaces.

We investigated the dynamics of thin-layer formation by non-spherical motile phytoplankton in time-dependent shear flow, building on the seminal work of Durham et al. (2009 Science vol. 323, pp. 1067–1070), on spherical microswimmers in time-independent flows. By solving the torque balance equation for a microswimmer, we found that the system is highly damped for body sizes smaller than $10^{-3}$ m, with initial rotational motion dissipating quickly. From this torque balance, we also derived the critical shear for ellipsoidal microswimmers, which we validated numerically. Simulations revealed that the peak density of microswimmers is slightly higher than the theoretical prediction due to the speed asymmetry of sinking and gyrotaxis above and below the predicted height. In addition, we observed that microswimmers with higher aspect ratios tend to form thicker layers due to slower angular velocity. Using linear stability analysis, we identified a thin-layer accumulation time scale, which contains two regimes. This theoretically predicted accumulation time scale was validated through simulations. In time-dependent flow with oscillating critical shear depth, we identified three accumulation regimes and a transitional regime based on the ratio of swimmer and flow time scales. Our results indicate that thin layers can form across time scale ratios spanning five orders of magnitude, which helps explain the widespread occurrence of thin phytoplankton layers in natural water bodies.

The effects of surface roughness in the transitionally rough regime on the overlying near-wall turbulence are modelled using quasi-linear approximations proposed recently: minimal quasi-linear approximation (MQLA) (Hwang & Ekchardt, 2020, J. Fluid Mech., vol. 894, A23), data-driven quasi-linear approximation (DQLA) (Holford et al., 2024, J. Fluid Mech., vol. 980, A12) and a newly established variant of MQLA (M2QLA, minimal two-mode quasi-linear approximation). The transpiration-resistance model (TRM) for boundary conditions is applied to account for the surface roughness (Lācis et al., 2020, J. Fluid Mech., vol. 884, A21). It is shown that many essential near-wall turbulence statistics are fairly well captured by the quasi-linear approximations in a wide range of slip and transpiration lengths for the TRM boundary conditions. In particular, the virtual origins and the resulting roughness functions are well predicted, showing good agreement with those from previous direct numerical simulations (DNS) in mild roughness cases. The DQLA and M2QLA, which incorporate streamwise-dependent Fourier modes in the approximations, are also shown to perform a little better than MQLA, especially with DQLA reproducing the two-dimensional energy spectra qualitatively consistent with the DNS. Finally, with a computational cost much lower than DNS, it is shown that the proposed quasi-linear approximation frameworks offer an efficient tool to rapidly explore the roughness effects within a large parameter space.

This study investigated the cylindrically divergent Rayleigh–Taylor instability (RTI) on a liquid–gas interface and its dependence on initial conditions. A novel hydrophobic technique was developed to generate a two-dimensional water–air interface with controlled initial conditions. The experimental configuration utilised high-pressure air injection to produce uniform circumferential acceleration. Amplitude measurements over time revealed that the cylindrical RTI growth depends strongly on the azimuthal wavenumber. Experimental results demonstrated that surface tension significantly suppresses the liquid–gas cylindrical RTI, even inducing a freeze-out and oscillatory perturbation growth – a phenomenon observed for the first time. Spectrum analysis of the interface contours demonstrated that the cylindrical RTI evolves in a weakly nonlinear regime. Linear and weakly nonlinear models were derived to accurately predict the time-varying interface amplitudes and high-order modes. The linear model was further used to determine conditions for unstable, freeze-out and oscillatory solutions of the cylindrically divergent RTI. These findings offer valuable insights into manipulating hydrodynamic instabilities in contracting/expanding geometries using surface tension.

The paper presents a simulation of the turbulent flow over and through a submerged aquatic canopy composed of 672 long, slender ribbons modelled as Cosserat rods. It is characterized by a bulk Reynolds number of 20 000, and a friction Reynolds number of 2638. Compared with a smooth turbulent channel at the same bulk Reynolds number, the canopy increases drag by a factor of 12. The ribbons are highly flexible, with a Cauchy number of 25 000, slightly buoyant, and densely packed. Their length exceeds the channel height by a factor of 1.6, while their average reconfigured height is only a quarter of the channel height. Different from lower-Cauchy-number cases, the movement of the ribbons, characterized by the motion of their tips, is very pronounced in the vertical direction, and even more in the spanwise direction, with root-mean-square fluctuations of the spanwise tip position 1.5 times the vertical ones. A canopy hull is defined to analyse the collective motion of the canopy and its interaction with the outer flow. Dominant spanwise wavelengths at this interface measure approximately one channel height, corresponding to twice the spacing of adjacent high- and low-speed streaks identified in two-point correlations of fluid velocity fluctuations. Conditional averages associated with troughs and ridges in the topography of the hull reveal streamwise-oriented counter-rotating vortices. They are reminiscent of the head-down structures related to the monami phenomenon in lower-Cauchy-number cases.

Predicting particle segregation has remained challenging due to the lack of a general model for the segregation velocity that is applicable across a range of granular flow geometries. Here, a segregation-velocity model for dense granular flows is developed by exploiting force balance and recent advances in particle-scale modelling of the segregation driving and drag forces over the entire particle concentration range, size ratios up to 3 and inertial numbers as large as 0.4. This model is shown to correctly predict particle segregation velocity in a diverse set of idealised and natural granular flow geometries simulated using the discrete element method. When incorporated in the well-established advection–diffusion–segregation formulation, the model has the potential to accurately capture segregation phenomena in many relevant industrial applications and geophysical settings.

Resolvent-based modelling and estimation is critically dependent on the nonlinear forcing input and hence understanding its role in the flow response is of great significance. This study quantifies the nonlinear forcing input in the resolvent formulation and investigates its characteristics for compressible turbulent boundary layers at Mach number 5.86 and friction Reynolds number 420 subject to adiabatic- and cold-wall conditions. Results show that, with the addition of the eddy viscosity to the resolvent operator, the cross-spectral density (CSD) of the forcing tends to exhibit a spatially uncorrelated distribution, which suggests that the spatial cross-coherence may be neglected and makes the modelling of the forcing input potentially easier. Aiming to quantify the different importance of each forcing component in generating turbulent fluctuations, contributions of the eddy-viscosity-corrected forcing to the flow responses are investigated through reduced-order analysis and matrix decomposition. The streamwise motions are almost insensitive to the temperature-related forcing, and can be oppositely influenced by the wall-normal and spanwise forcing components. By retaining only the diagonal components in the CSD of the forcing input, the assumption of forcing decorrelation in space and among components is also examined in the input–output framework. It is found that this simplified input is able to capture the dominant turbulence features and the local forcing is observed to cause inner-layer responses. That is, present results suggest adequate modelling of the CSD of the forcing can be achieved retaining only its diagonal components. On the basis of the current findings, the forcing input in the resolvent-based framework is thus modelled, with the wall-normal dependence and amplitude ratio between forcing components designed for compressible turbulent boundary layers. Through an algebraic Lyapunov equation, improved estimations of the statistical spectral densities of velocity and temperature fluctuations are finally obtained, in contrast to the results by simply assuming the forcing CSD to be an identity matrix.

Direct numerical simulations are carried out to investigate the underlying mechanism of the low-frequency unsteadiness of a transitional shock reflection with separation at $M=1.5$. To clarify the nonlinear mechanisms, the incoming laminar boundary layer is forced with two different arrangements of oblique unstable modes. Each wave arrangement is given by a combination of two unstable waves such that their difference in frequency falls in a low-frequency range corresponding to a Strouhal number (based on the length of interaction) of 0.04. This deterministic forcing allows the introduction of nonlinearities, and high-order statistical tools are used to identify the properties of quadratic couplings. It is found that the low-frequency unsteadiness and the transition to turbulence are decoupled problems. On the one hand, the unstable modes of the boundary layer interact nonlinearly such that energy cascades to higher frequencies, initiating the turbulent cascade process, and to lower frequencies. On the other hand, the low-frequency quadratic coupling of the oblique modes is found to be responsible for low-frequency unsteadiness affecting the separation point. The direction of the quadratic interactions is extracted and it is shown that, in the presence of low-frequency unsteadiness, these interactions enter the separated zone just before reattachment and travel both downstream and upstream, extending beyond the separation point, hence feeding the low-frequency bubble response. In addition to the two main arrangements of oblique modes, two other combinations are analysed, including multiple oblique waves and streaks. Interestingly, their inclusion did not alter the low-frequency unsteadiness phenomenon. Furthermore, the effect of the forcing difference frequency is examined and it is shown that the breathing phenomenon is sensitive to the range of frequencies present in the system due to a low-pass filter effect.

Mass transport induced by group-forced subharmonic waves (infragravity waves) is investigated in the present study. A theoretical solution for subharmonic waves’ kinematic contributions to fourth-order mass transport and drift velocity has been proposed for any depth and bandwidth for the first time. This model is validated using particle-tracking simulations driven by the flow field generated by the SWASH. The subharmonic-induced mass transport solution is a weighted sum of the subharmonic velocity variance spectrum and velocity skewness bispectrum due to the triad-difference interaction among two primary and one subharmonic components. For narrow-banded waves with long wave group relative to depth, the weightings become independent of spectral components, and the solution is recovered in the time domain. Two mechanisms contributing to mass transport were identified: a forward drift resulting from self-interaction similar to Stokes drift, and a depth-decaying backward drift induced by negative subharmonic velocity skewness due to the anti-phase coupling between subharmonics and wave groups. For narrow-banded waves the forward transport surpasses the backward transport for kh< 0.72, where k is the short wave wavenumber and h is the water depth. For other waves, the critical kh for this phenomenon decreases with increasing wave period and bed slope and decreasing bandwidth. At greater depths or steeper bed slopes, near-surface backward transport predominates over forward transport; at shallower depths or gentler slopes, forward transport is dominant throughout the water column. Although smaller than Stokes transport by short waves, the subharmonic wave-induced mass transport can affect the long-term trajectory of a floating and suspended particle. This study provides the first evidence and insight for the influences of group-forced subharmonics on vertically varying mass transport from the ocean surface to seabed in coastal environments.

Convection in planetary environments is often modelled using stress-free boundary conditions, with diffusion-free geostrophic turbulence scalings frequently assumed. However, key questions remain about whether rotating convection with stress-free boundary conditions truly achieves the diffusion-free geostrophic turbulence regime. Here, we investigated the scaling behaviours of the Nusselt number ($Nu$), Reynolds number (${Re}$) and dimensionless convective length scale ($\ell /H$, where $H$ is the height of the domain) in rotating Rayleigh–Bénard convection under stress-free boundary conditions within a Boussinesq framework. Using direct numerical simulation data for Ekman number $Ek$ down to $5\times 10^{-8}$, Rayleigh number $Ra$ up to $5\times 10^{12}$, and Prandtl number $Pr = 1$, we show that the diffusion-free scaling of the heat transfer $Nu - 1 \sim Ra^{3/2}\, Pr^{-1/2}\, Ek^2$ alone does not necessarily imply that the flow is in a geostrophic turbulence regime. Under the stress-free conditions, ${Re}$ and $\ell /H$ deviate from the diffusion-free scalings, indicating a dependence on molecular diffusivity. We propose new non-diffusion-free scaling relations for this diffusion-free heat transfer regime with stress-free boundary conditions: $\ell /H \sim Ra^{1/8}\, Pr^{-1/8}\, Ek^{1/2}$ and ${Re} \sim Ra^{11/8}\, Pr^{-11/8}\, Ek^{3/2}$. Our findings highlight the need to assess both thermal and dynamic characteristics to confirm geostrophic turbulence.

In turbulent pipe flows, drag-reducing polymers are commonly used to reduce skin-friction drag; however, predicting this reduction in industry applications, such as crude oil pipelines, remains challenging. The skin-friction coefficient ($C_f$) of polymer drag-reduced turbulent pipe flows can be related to three dimensionless parameters: the solvent Reynolds number ($Re_s$), the Weissenberg number ($Wi$) and the ratio of solvent viscosity ($\eta _s$) to zero-shear-rate viscosity ($\eta _0$), denoted as $\beta$. The function that relates these four dimensionless numbers was determined using experiments of various pipe diameters ($D$), flow velocities ($U$) and drag-reducing polyacrylamide solutions. The experiments included measurements of streamwise pressure drop ($\Delta P$) for determining $C_f$, and measurements of shear viscosity ($\eta$) and elastic relaxation time ($\lambda$). This experimental campaign involved 156 flow conditions, each characterised by distinct values for $C_f$, $Re_s$, $Wi$ and $\beta$. Experimental results demonstrated good agreement with the relationship: $C_f^{-1/2} = \widehat {A}\log _{10}(Re_sC_f^{1/2})+\widehat {B}$, where $\widehat {A} = 27.6(Wi \beta )^{0.346}$ and $\widehat {B} = 122/15-58.9(Wi \beta )^{0.346}$. Based on this relationship, onset and maximum drag reduction are predicted to occur when $Wi \beta$ equals $3.76 \times 10^{-3}$ and $3.40 \times 10^{-1}$, respectively. This function can predict $C_f$ of dilute polyacrylamide solutions based on predefined parameters (bulk velocity, pipe diameter, density, solvent viscosity) and two measurable rheological properties of the solution (shear viscosity and elastic relaxation time) with an accuracy of $\pm 9.36$ %.

Turbulence closures are essential for predictive fluid flow simulations in both natural and engineering systems. While machine learning offers promising avenues, existing data-driven turbulence models often fail to generalise beyond their training datasets. This study identifies the root cause of this limitation as the conflation of generalisable flow physics and dataset-specific behaviours. We address this challenge using symbolic regression, which yields interpretable, white-box expressions. By decomposing the learned corrections into inner-layer, outer-layer and pressure-gradient components, we isolate universal physics from flow-specific features. The model is trained progressively using high-fidelity datasets for plane channel flows, zero-pressure-gradient turbulent boundary layers (ZPGTBLs), and adverse pressure-gradient turbulent boundary layers (PGTBLs). For example, direct application of a model trained on channel flow data to ZPGTBLs results in incorrect skin friction predictions. However, when only the generalisable inner-layer component is retained and combined with an outer-layer correction specific to ZPGTBLs, predictions improve significantly. Similarly, a pressure-gradient correction derived from PGTBL data enables accurate modelling of aerofoil flows with both favourable and adverse pressure gradients. The resulting symbolic corrections are compact, interpretable, and generalise across configurations – including unseen geometries such as aerofoils and Reynolds numbers outside the training set. The models outperform baseline Reynolds-averaged Navier–Stokes closures (e.g. the Spalart–Allmaras and shear stress transport models) in both a priori and a posteriori tests. These results demonstrate that explicit identification and retention of generalisable components is key to overcoming the generalisation challenge in machine-learned turbulence closures.

Compressible jets impinging on a perpendicular surface can produce high-intensity, discrete-frequency tones. The character of these tones is a function of nozzle shape, jet Mach number, impingement-plate geometry, and the distance between nozzle and plate. Though it has long been recognised that these tones are associated with a resonance cycle, the exact mechanism by which they are generated has remained a topic of some debate. In this work, we present evidence for a number of distinct tone-generation mechanisms, reconciling some of the different findings of prior authors. We demonstrate that the upstream-propagating waves that close resonance can be confined within the jet, or external to it. These waves can be either weak and relatively linear, or strong and nonlinear from their inception. The waves can undergo coalescence or merging, and in some configurations, pairs of waves rather than singletons appear. We discuss both historical and new evidence for multiple distinct processes by which upstream-propagating waves are produced: direct vortex sound, shock leakage, wall-jet-boundary fluctuations, and wall-jet shocklets. We link these various mechanisms to the disparate collection of upstream-propagating waves observed in the data. We also demonstrate that multiple mechanisms can be provoked by a single vortex, providing an explanation as to why sometimes pairs of waves or merging waves are observed. Through this body of work, we demonstrate that rather than being in opposition, the various pieces of past research on this topic were simply identifying different mechanisms that can support resonance.