A model is proposed for the one-dimensional spectrum and streamwise Reynolds stress in pipe flow for arbitrarily large Reynolds numbers. Constructed in wavenumber space, the model comprises four principal contributions to the spectrum: streaks, large-scale motions, very-large-scale motions and incoherent turbulence. It accounts for the broad and overlapping spectral content of these contributions from different eddy types. The model reproduces well the broad structure of the premultiplied one-dimensional spectrum of the streamwise velocity, although the bimodal shape that has been observed at certain wall-normal locations, and the $-5/3$ slope of the inertial subrange, are not captured effectively because of the simplifications made within the model. Regardless, the Reynolds stress distribution is well reproduced, even within the near-wall region, including key features of wall-bounded flows such as the Reynolds number dependence of the inner peak, the formation of a logarithmic region, and the formation of an outer peak. These findings suggest that many of these features arise from the overlap of energy content produced by both inner- and outer-scaled eddy structures combined with the viscous-scaled influence of the wall. The model is also used to compare with canonical turbulent boundary layer and channel flows, and despite some differences being apparent, we speculate that with only minor modifications to its coefficients, the model can be adapted to these flows as well.

Yaw control can effectively enhance wind farm power output, but the vorticity distribution and coherent structures in yawed turbine wakes remain poorly understood. We propose a physical model capable of accurately predicting tip vortex dynamics from their generation to destabilisation. This model integrates a point vortex framework with advanced blade element momentum theory and vortex cylinder theory for yawed turbines. Comparisons with large eddy simulations demonstrate that the model effectively predicts the vorticity distribution of tip vortices and the wake profile of yawed turbines. Finally, we employ sparsity-promoting dynamic mode decomposition to analyse the dynamics of the far wake. Our analysis reveals four primary mode types: (i) the averaged mode; (ii) shear modes; (iii) harmonic modes; and (iv) merging modes. Under yawed conditions, these modes become asymmetric, leading to interactions between the tip and root vortex modes. This direct interaction plays a critical role during the formation process of the counter-rotating vortex pair observed in yawed wakes.

Linear hydroelastic waves propagating in a frozen channel are investigated. The channel has a rectangular cross-section, finite depth and infinite length. The liquid in the channel is an inviscid, incompressible liquid and covered with ice. The ice is modelled as a thin elastic plate of variable thickness clamped to the channel walls. The thickness is constant along the channel length and varying across it. The flow induced by ice deflections is potential. The problem reduced to a problem of wave profiles across the channel and was solved using a piecewise linear approximation of a shape of the thickness. Normal modes are calculated to ensure continuous deflections, slopes, bending stresses and shear forces in the ice plate. Two thickness distributions are studied: in Case I, the thickness is constant at a middle segment and linearly increases at edge segments over the channel’s width; in Case II, the thickness linearly decreases at the edge segments. In Case I, there is one segment with the thin part of the ice cover where, as expected, the oscillations of the ice plate will be concentrated. In Case II, there are two such areas, separated by the middle segment with the thick part of the ice cover. Dispersion relations, phase and group velocities, wave profiles and strain distributions in the ice plate are studied. Results show that the properties of periodic hydroelastic waves are significantly influenced by the ice thickness distribution across the channel.

We investigate the effect of streamwise and transverse rotation on the wake behind an elastically mounted sphere. Simulations are performed at a Reynolds number $Re=500$ over a range of reduced velocity $2\le U^{\ast }\le 12$, considering a low and high rotational speed (0.2 and 1), keeping the mass ratio $m^{\ast }=2$. Streamwise rotation yields a structural response akin to the non-rotating case, while transverse rotation triggers induced vibration at lower $U^{\ast }$ and sustains it across a wider range. Like the non-rotating case, the streamwise rotating sphere exhibits synchronous, high-amplitude vibration across the entire $U^\ast$ range, whereas for low transverse rotation, it is confined to $5\le U^{\ast }\le 6$. Cross-stream displacement of the sphere remains unaffected by streamwise rotation with increasing $U^{\ast }$. In contrast, it monotonically increases due to transverse rotation, driven by the Magnus force, as supported by our theoretical and numerical estimations. While the spiral shedding mode dominates at $\Omega _{x}=0.2$, twisted hairpin and twisted spiral modes emerge as the rotation rate is increased. On the other hand, we observe the hairpin (HP) mode, as seen in the non-rotating case, for low transverse rotation. The HP mode gives rise to the ring vortical mode at the far wake, and with an increase in $U^\ast$, the wake shows small-scale stretched threads and reconnected bridgelets. Wake fluctuations increase with a streamwise rotation that saturates at higher $U^{\ast }$ during synchronisation, while desynchronisation at dimensionless transverse rotation rate $\Omega _{z}=1$ induces intermittent low-amplitude vibration via the Magnus effect. Space–time reconstruction at the near wake shows an undisturbed helical vortex core at $\Omega _{x}=0.2$ and $U^{\ast }=5$, which bifurcates at $\Omega _{x}=1$ owing to the centrifugal-induced distortion. At $\Omega _{x}=1$ and $U^{\ast }=5$, the phase difference between $(y, C_{y})$ and $(z, C_{z})$ exhibits in-phase synchrony with occasional phase slips. The wake vortex remains unaffected by the transverse rotation of the sphere; however, a streamwise rotating sphere couples the wake, leading to a rotational lock-in. The wake rotation shifts from anti-clockwise to clockwise sense earlier (in $U^\ast$) at a lower rotation rate. The reduced velocity is seen to have a favourable effect on the transfer of the sphere’s rotational inertia onto the wake as the measured penetration depth increases with $U^{\ast }$. Insights from the present research will aid in understanding complex flow interactions in rotational systems, improving efficiency, stability and control in modern engineering applications.

Phoretic particles are often used as a simple model for experimental and theoretical studies of active matter. We develop a computational framework to resolve hydrodynamic and chemical interactions of multiple self-propelling phoretic particles suspended in two-dimensional Stokes flow. The proposed method is precise enough to resolve correctly the subtle transitions between different modes of spontaneous locomotion for a single particle, and fast and versatile enough to study multiparticle dynamics in periodic or confined domains. The particles are modelled as chemically active rigid circles, which can emit or absorb a solute into surrounding fluid. The interaction between particles and solute induces a slip flow on particle surfaces, and the solute is advected by the fluid flow and diffuses with a constant diffusivity. A fast boundary integral method is proposed to solve fluid–structure interaction in Stokes flow. Acceleration of this method is provided by splitting the velocity field due to a set of point forces into a short-range part with singularity and a long-range part which is sufficiently smooth, thanks to an Ewald-like decomposition. An overlapping mesh method is employed for advection–diffusion of the solute with moving boundaries. The idea is to decompose the computational domain into several overlapping subdomains, and body-fitted meshes are used to ensure sharp resolution of boundary conditions. The framework is validated separately for the Stokes problem and the advection–diffusion problem, reaching relatively high order of accuracy. We apply our framework to several practical problems, such as a single particle in a channel and particle suspensions, showing rich sets of behaviours.

The relationship between salinity-driven (SD) and particle-driven (PD) gravity currents has long been a focal point of geophysical research. This study investigates salinity–particle dual-driven gravity currents using a direct numerical simulation discrete element method. The transition regime from SD to PD currents is explored. The results show that the transition is related to interfacial instability and material transport dynamics. During this transition, the enhancement of particle sedimentation weakens the interfacial stratification and heightens its susceptibility to shear instability. Consequently, the instability generates a series of billows that encourages fluid dilution, further amplifying the particle sedimentation effect. The transition regime is closely associated with this positive closed-loop feedback mechanism. It supplies sufficient energy at the slumping stage to maintain the front velocity of particle-dominated currents comparable to that of salinity-dominated currents. The interfacial vortices will expand spatially by the centrifugal forces on the particles, leading to a reduction in detrainment.

Fourier analysis is the standard tool of choice for quantifying the distribution of kinetic energy amongst the eddies in a turbulent flow. The resulting spectral energy-density function is the well-known energy spectrum. And yet, because eddies are distinct from waves, alternative approaches to finding energy-density functions have long been sought. Townsend (1976) outlined a promising approach to finding a spatial energy-density function, $V\!(r)$, where $r$ is the eddy size. Notably, this approach led to two distinct and mutually inconsistent formulations of $V\!(r)$ in homogeneous, isotropic turbulence. We revisit Townsend’s proposal and derive the corresponding three-dimensional $V\!(r)$ as well as introduce its one-dimensional variants (which, to our knowledge, have not been explicitly discussed before). By training our focus on the associated dimensionality of the function, we resolve the discrepancies between the previous formulations. Additionally, we generalise our analysis to include anisotropic flows. Finally, by means of concrete examples, we illustrate how one-dimensional spatial energy-density functions are useful for analysing empirical data. Some notable findings include new insights into the $k_1^{-1}$ scaling (where $k_1$ is the streamwise wavenumber) and a possible resolution of the enigmatic sizes of organised motions at large scales.

Shock interactions on a V-shaped blunt leading edge (VBLE) that are commonly encountered at the cowl lip of an inward-turning inlet are investigated at freestream Mach numbers ($ M_\infty$) 3–6. The swept blunt leading edges of the VBLE generate a pair of detached shocks with varying shapes due to the changes in $ M_\infty$ and $L/r$ (i.e. the ratio of the leading-edge length $L$ to the leading-edge blunt radius $r$), which causes intriguing shock interactions at the crotch of the VBLE. Three subtypes of regular reflection (RR) and a Mach reflection (MR) are produced successively with increasing $ M_\infty$ for a given $L/r$, which appear in the opposite order to those with increasing $L/r$ for a given $ M_\infty$. These shock interactions identified in numerical simulations are verified in supersonic and hypersonic wind tunnel experiments. It is demonstrated that the relative position of the shocks is crucial in determining the transitions of shock interactions by varying either $L/r$ or $ M_\infty$. Transition criteria between subtypes of RR and from RR to MR are theoretically established in the parameter space $(M_\infty,L/r)$ by analysing the shock structures, showing good agreement with the numerical and experimental results. Interactions between either immature or fully developed detached shocks are embedded in these criteria. Specifically, the transition criteria asymptotically approach the corresponding critical $ M_\infty$ when $L/r$ is sufficiently large. These transition criteria provide guidelines for improving the design of the cowl lip of an inward-turning inlet in supersonic and hypersonic regimes.

Experiments were performed that (i) document the effect of the steady spanwise buffer layer blowing on the mean characteristics of the turbulent boundary layer for a range of momentum thickness Reynolds numbers from 4760 to 10 386, and (ii) document the effect of the buffer layer blowing on the unsteady characteristics and coherent vorticity in a boundary layer designed to provide sufficiently high spatial resolution. The spanwise buffer layer blowing of the order of $u_{\tau }$ is produced by a surface array of pulsating direct current (pulsed-DC) plasma actuators. This was found to substantially reduce the wall shear stress that was directly measured with a floating element coupled with a force sensor. The direct wall shear measurements agreed with values derived using the Clauser method to within $\pm 0.85$ %. The degree to which the buffer layer blowing affected $\tau _w$ was found to primarily depend on the inner variable spanwise spacing between the pulsed-DC actuator electrodes, i.e. ‘blowing sites’. Utilizing pairs of $[u,v]$ and $[u,w]$ hot-wire sensors, the latter experiments correlated significant reductions in the $\omega _y$ and $\omega _x$ vorticity components that resulted from the buffer layer blowing and translated into lower Reynolds stresses and turbulence production. The time scale to which these observed changes in the boundary layer characteristics would return to the baseline condition was subsequently documented. This revealed a recovery length of $x^+ \approx 86\,000$ that translated to a streamwise fetch of $x \approx 66\delta$. Finally, a comparison with the recent work by Cheng et al. (2021, J. Fluid Mech. vol. 918, A24) and Wei & Zhou (2024 in TSFP13, June 25–28, 2024) that followed our experimental approach to achieve comparable wall shear stress (drag) reductions has led to a new scaling based on the baseline boundary layer $\textit{Re}_{\tau }$ and buffer layer blowing velocity.

The Taylor–Melcher leaky dielectric (LD) model is often used to study the physics of electrosprays operating in the cone-jet mode. Despite its success, there are electrospraying conditions in which the ion concentration fields must be retained, which requires an electrokinetic model. This article reproduces cone-jets with two electrokinetic formulations: the standard Poisson–Nernst–Planck (PNP) equations, and a modified electrokinetic (MEK) model that accounts for overscreening and overcrowding of electrolytes, which is important in fluids with high electrical conductivities such as ionic liquids (Kilic et al. 2007 Phys. Rev. E vol. 75, no. 2, 021502, 021503; Bazant et al. 2011 Phys. Rev. Lett. vol. 106, no. 4, 46102). In the case of liquids with low electrical conductivities, it is observed that the LD and PNP models agree under certain limiting conditions, but they are less restrictive than previously proposed (Baygents & Saville 1990 AIP Conf. Proc. vol. 197, 7–17; Schnitzer & Yariv 2015 Fluid Mech. vol. 773, 1–33); the effects of dissimilar ion diffusivities are also investigated. In the case of liquids with high electrical conductivities, in particular ionic liquids, overscreening and overcrowding effects are important, resulting in significant differences between the solutions of the PNP, MEK and LD models. In particular, the electrokinetic models yield increased dissipation and self-heating, leading to higher temperature variations and currents, in agreement with measurements. Furthermore, the MEK formulation describes the ion concentration fields with higher fidelity than the PNP equations.

In this study, we demonstrate, for the first time, the existence of a short-wave instability in a Lamb–Oseen vortex subjected to a triangular strain field generated by three satellite vortices, which we term the triangular instability. We identify this instability by numerically integrating the linearised Navier–Stokes equations around a quasi-steady base flow to capture the most unstable mode and validate it by comparing results with theoretical predictions. We evaluate this instability by calculating the growth rates associated with the parametric resonant coupling of two Kelvin waves with the triangular strain field in the limit of small strain rate and large Reynolds number. Our analysis reveals that resonance occurs only for combinations of the azimuthal wavenumbers $m = 1$ and $m = - 2$ (or their symmetric counterparts with opposite signs). We observe several unstable modes with positive growth rates for a moderate viscous Reynolds number $10^4$ and straining parameter value $\epsilon = 0.008$, defined as the cube of the ratio of the core size to the distance from the satellite vortices. The most unstable mode, dominant at typically high Reynolds numbers, has $k \approx 5.18/a$ and $\omega \approx - 0.312\Omega$ (where $a$ and $\Omega$ denote the core size and central angular velocity). It exhibits negligible critical layer damping and remains the most unstable mode over a wide range of ${Re}$ and $\epsilon$. At lower Reynolds numbers, another mode with $k \approx 1.76/a$ and $\omega \approx - 0.407\Omega$, despite significant critical layer damping, becomes the most unstable.

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.