A phenomenological description is presented to explain the intermediate and low-frequency/large-scale contributions to the wall-shear-stress (${\tau }_w$) and wall-pressure ($\,{p}_w$) spectra of canonical turbulent boundary layers, both of which are well known to increase with Reynolds number, albeit in a distinct manner. The explanation is based on the concept of active and inactive motions (Townsend, J. Fluid Mech., vol. 11, issue 1, 1961, pp. 97–120) associated with the attached-eddy hypothesis. Unique data sets of simultaneously acquired ${\tau }_w$, ${p}_w$ and velocity-fluctuation time series in the log region are considered, across a friction-Reynolds-number ($Re_{\tau }$) range of $ {O}(10^3) \lesssim Re_{\tau } \lesssim {O}(10^6)$. A recently proposed energy-decomposition methodology (Deshpande et al., J. Fluid Mech., vol. 914, 2021, A5) is implemented to reveal the active and inactive contributions to the ${\tau }_w$- and $p_w$-spectra. Empirical evidence is provided in support of Bradshaw's (J. Fluid Mech., vol. 30, issue 2, 1967, pp. 241–258) hypothesis that the inactive motions are responsible for the non-local wall-ward transport of the large-scale inertia-dominated energy, which is produced in the log region by active motions. This explains the large-scale signatures in the ${\tau }_w$-spectrum, which grow with $Re_{\tau }$ despite the statistically weak signature of large-scale turbulence production, in the near-wall region. For wall pressure, active and inactive motions respectively contribute to the intermediate and large scales of the $p_w$-spectrum. Both these contributions are found to increase with increasing $Re_{\tau }$ owing to the broadening and energization of the wall-scaled (attached) eddy hierarchy. This potentially explains the rapid $Re_{\tau }$-growth of the $p_w$-spectra relative to ${\tau }_w$, given the dependence of the latter only on the inactive contributions.

This study employs direct numerical simulations to examine the effects of varying backpressure conditions on the turbulent atomisation of impinging liquid jets. Using the incompressible Navier–Stokes equations, and a volume-of-fluid approach enhanced by adaptive mesh refinement and an isoface-based interface reconstruction algorithm, we analyse spray characteristics in the environments with ambient gas densities ranging from 1 to 40 times the atmospheric pressure under five different backpressure scenarios. We investigate the behaviour of turbulent jets, incorporate realistic orifice geometries and identify significant variations in the atomisation patterns depending on backpressure. Two distinct atomisation types emerge, namely jet-sheet-ligament-droplet at lower backpressures and jet-sheet-fragment-droplet at higher ones, alongside a transition from dilute to dense spray patterns. This variation affects the droplet size distribution and spray dynamics, with increased backpressure reducing the spray's spreading angle and breakup length, while increasing the droplet size variation. Furthermore, these conditions promote distributions that induce rapid, nonlinear wavy motion in liquid sheets. Topological analysis of the atomisation field using velocity-gradient tensor invariants reveals significant variations in topology volume fractions across different regions. Downstream, the droplet Sauter mean diameter increases and then stabilises, reflecting the continuous breakup and coalescence processes, notably under higher backpressures. This research underscores the substantial impact of backpressure on impinging-jet atomisation and provides essential insights for nozzle design to optimise droplet distributions.

We have investigated the dynamics of floating tracer in an idealised turbulent quasi-geostrophic ocean by advecting Lagrangian particles in a high-resolution velocity field enhanced by the potential flow associated with vortex stretching. At first order in the Rossby number expansion, this component of the ageostrophic circulation can be derived through a diagnostic equation in terms of the geostrophic velocities. Borrowing methods from the theory of Lagrangian coherent structures, we identify coherent material loops around strong vortex cores using the Lagrangian averaged vorticity deviation (LAVD). Building on studies of clustering in kinematic, stochastic velocity fields, we utilise methods from statistical topography to show that the coherent vortices dominate the distribution of extreme values of the concentration field. We find that the presence of clusters and voids in a coherent vortex depends on more than just the sense of rotation, but also on the full evolution of the vorticity over its lifecycle. We identify the mechanism behind the cluster formation that respects the symmetries of the quasi-geostrophic equations but can be expected to hold robustly in more complicated regimes, due to the simple physical description. The association of cluster formation with vortex stretching implies that LAVD is a particularly relevant metric for floating tracer dynamics. The detection of intense clustering also has implications for reaction rates between ocean-borne flotsam, meaning that our results are relevant to understanding the general risk of floating microplastics and marine biological populations.

We develop a time-dependent conformal method to study the effect of viscosity on steep surface waves. When the effect of surface tension is included, numerical solutions are found that contain highly oscillatory parasitic capillary ripples. These small-amplitude ripples are associated with the high curvature at the crest of the underlying viscous-gravity wave, and display asymmetry about the wave crest. Previous inviscid studies of steep surface waves have calculated intricate bifurcation structures that appear for small surface tension. We show numerically that viscosity suppresses these. While the discrete solution branches still appear, they collapse to form a single smooth branch in the limit of small surface tension. These solutions are shown to be temporally stable, both to small superharmonic perturbations in a linear stability analysis, and to some larger amplitude perturbations in different initial-value problems. Our work provides a convenient method for the numerical computation and analysis of water waves with viscosity, without evaluating the free-boundary problem for the full Navier–Stokes equations, which becomes increasingly challenging at larger Reynolds numbers.

Gravito–capillary waves at free surfaces are ubiquitous in several natural and industrial processes involving quiescent liquid pools bounded by cylindrical walls. These waves emanate from the relaxation of initial interface distortions, which often take the form of a cavity (depression) centred on the symmetry axis of the container. The surface waves reflect from the container walls leading to a radially inward propagating wavetrain converging (focussing) onto the symmetry axis. Under the inviscid approximation and for sufficiently shallow cavities, the relaxation is well-described by the linearised potential-flow equations. Naturally, adding viscosity to such a system introduces viscous dissipation that enervates energy and dampens the oscillations at the symmetry axis. However, for viscous liquids and deeper cavities, these equations are qualitatively inaccurate. In this study, we decompose the initial localised interface distortion into several Bessel functions and study their time evolution governing the propagation of concentric gravito–capillary waves on a free surface. This is carried out for inviscid as well as viscous liquids. For a sufficiently deep cavity, the inward focussing of waves results in large interfacial oscillations at the axis, necessitating a second-order nonlinear theory. We demonstrate that this theory effectively models the interfacial behaviour and highlights the crucial role of nonlinearity near the symmetry axis. This is rationalised via demonstration of the contribution of bound wave components to the interface displacement at the symmetry axis Contrary to expectations, the addition of slight viscosity further intensifies the oscillations at the symmetry axis although the mechanism of wavetrain generation here is quite different compared with bubble bursting where such behaviour is well known (Duchemin et al., Phys. Fluids, vol. 14, issue 9, 2002, pp. 3000–3008). This finding underscores the limitations of the potential flow model and suggests avenues for more accurate modelling of such complex free-surface flows.

Aircraft with bio-inspired flapping wings that are operated in low-density atmospheric environments encounter unique challenges associated with the low density. The low density results in the requirement of high operating velocities of aircraft to generate sufficient lift resulting in significant compressibility effects. Here, we perform numerical simulations to investigate the compressibility effects on the lift generation of a bio-inspired wing during hovering flight using an immersed boundary method. The aim of this study is to develop a scaling law to understand how the lift is influenced by the Reynolds and Mach numbers, and the associated flow physics. Our simulations have identified a critical Mach number of approximately $0.6$ defined by the average wing-tip velocity. When the Mach number is lower than 0.6, compressibility does not have significant effects on the lift or flow fields, while when the Mach number is greater than $0.6$, the lift coefficient decreases linearly with increasing Mach number, due to the drastic change in the pressure on the wing surface caused by unsteady shock waves. Moreover, the decay rate is dependent on the Reynolds number and the angle of attack. Based on these observations, we propose a scaling law for the lift of a hovering flapping wing by considering compressible and viscous effects, with the scaled lift showing excellent collapse.

Models for slow flow of dense granular materials often treat the medium as incompressible, thereby neglecting the role of Reynolds dilatancy. However, recent particle simulations have demonstrated the presence of a significant coupling between the volume fraction and velocity fields. The model of Dsouza & Nott (J. Fluid Mech., vol. 888, 2020, R3) incorporates dilatancy and captures the coupling, but it has thus far lacked experimental validation. In this paper, we provide the first experimental demonstration of dilatancy and its coupling to the kinematics in a two-dimensional cylindrical Couette cell. We find a shear layer near the inner cylinder within which there is significant dilation. Within the shear layer, the azimuthal velocity decays roughly exponentially and the volume fraction rises with radial distance from the inner cylinder. The predictions of the model of Dsouza & Nott (2020) are in good agreement with the experimental data for a variety of roughness features of the outer cylinder. Moreover, by comparing the steady states resulting from different initial volume fraction profiles (but having the same average), we show the inter-dependence of the velocity and volume fraction fields, as predicted by the model. Our results establish the importance of shear dilatancy even in systems of constant volume.

In this study, we investigate the properties of energy thickness $\delta _3$ in turbulent boundary layer (TBL) flows, a parameter derived solely from the mean streamwise velocity ($U$) profile. Through an analysis of the energy integral equation for zero pressure gradient TBLs, we establish a close relationship between turbulent kinetic energy (TKE) production and $\delta _3$, offering a practical method to estimate TKE production, which is particularly useful in physical experiments where direct measurements are challenging. The significance of $\delta _3$ becomes even more pronounced in TBLs under pressure gradient. Through extensive analysis of numerical and experimental data, we show that the ratio between $\delta _3$ and the momentum thickness $\delta _2$ is a promising criterion for predicting flow separation. Moreover, we derive a new energy integral equation for TBLs under arbitrary pressure gradients, and provide approximations for TKE productions terms by $R_{uv}\,\partial U/\partial y$ and $R_{uu}\,\partial U/\partial x$, and dissipation term by the mean shear. Here, $x, y$ represent the streamwise and wall-normal directions, respectively, and $R_{uu}$ and $R_{uv}$ are the Reynolds normal and shear stresses. The accuracy and robustness of the new energy integral equation and the approximation equations are validated using direct numerical simulations data. Our results show that the TKE production by $R_{uv}\,\partial U/\partial y$ and the overall productions consistently remain positive, reflecting a continuous conversion of mean kinetic energy into TKE across all TBLs. However, under strong favourable pressure gradients, TKE production by $R_{uu}\,\partial U/\partial x$ becomes negative, indicating a reverse energy transfer from TKE to mean kinetic energy.

Predicting and perhaps mitigating against rare, extreme events in fluid flows is an important challenge. Due to the time-localised nature of these events, Fourier-based methods prove inefficient in capturing them. Instead, this paper uses wavelet-based methods to understand the underlying patterns in a forced flow over a 2-torus which has intermittent high-energy burst events interrupting an ambient low-energy ‘quiet’ flow. Two wavelet-based methods are examined to predict burst events: (i) a wavelet proper orthogonal decomposition (WPOD) based method which uncovers and utilises the key flow patterns seen in the quiet regions and the bursting episodes; and (ii) a wavelet resolvent analysis (WRA) based method that relies on the forcing structures which amplify the underlying flow patterns. These methods are compared with a straightforward energy tracking approach which acts as a benchmark. Both the wavelet-based approaches succeed in producing better predictions than a simple energy criterion, i.e. earlier prediction times and/or fewer false positives and the WRA-based technique always performs better than WPOD. However, the improvement of WRA over WPOD is not as substantial as anticipated. We conjecture that this is because the mechanism for the bursts in the flow studied is found to be largely modal, associated with the unstable eigenfunction of the Navier–Stokes operator linearised around the mean flow. The WRA approach should deliver much better improvement over the WPOD approach for generically non-modal bursting mechanisms where there is a lag between the imposed forcing and the final response pattern.

The three-dimensional stability of two-dimensional natural convection flows in a heated, square enclosure inclined to the horizontal is investigated numerically. First, the computational procedure is validated by comparison of base flow solutions to results reported in literature across a range of inclinations. A bi-global linear stability analysis is then conducted to investigate the stability of these two-dimensional base flows to infinitesimal three-dimensional perturbations, and the effect that buoyancy forces (defined by a buoyancy number $R_N$) and enclosure inclination $\theta$ have on these stability characteristics. The flow is first observed to become three-dimensionally unstable at buoyancy number $R_N = 213.8$ when $\theta$ is $180^\circ$; this increases to $R_N = 2.54 \times 10^4$ at inclination $\theta =58^\circ$. It is found that the two-dimensional base flow is more unstable to three-dimensional perturbations with the critical $R_N$ corresponding to three-dimensional instability being significantly lower than its two-dimensional counterpart across all considered inclinations except $83^\circ \leq \theta \leq 88^\circ$, where the most unstable mode is a two-dimensional oscillatory mode that develops in the boundary layers along the conducting walls. Eight different leading three-dimensional instability modes are identified, with inclinations $58^\circ \leq \theta < 88^\circ$ transitioning through an oscillatory mode, and inclinations $88^\circ \leq \theta \leq 180^\circ$ transitioning through a stationary mode. The characteristics of the primary instability modes corresponding to inclinations $88^\circ \leq \theta \leq 179^\circ$ indicate the presence of a Taylor–Görtler instability.

We undertake an experimental investigation into the instabilities that emerge when a shear-thinning fluid intrudes a less viscous Newtonian fluid axisymmetrically in a lubricated Hele-Shaw cell. Pre-formed lubrication layers of Newtonian fluid that separate the shear-thinning fluid from the cell walls are incorporated into the experimental design. Provided the lubrication layers remain effective at reducing shear stress, so that extensional stresses dominate the flow of the intruding fluid, the instabilities evolve to form branch-like structures, which exhibit fracturing or tearing behaviour at their troughs. Thicker lubrication layers enable the branches to propagate radially outwards, whilst thinner, less effective ones hinder their development and progression. In the absence of lubrication layers, the shear-thinning fluid spreads radially and remains axisymmetric. For lubricated flows, we show that the number of branches is dependent primarily on the strain rate at the radial distance where they first emerge, and that the number of branches decreases with increasing strain rate.

The manipulation of the Richtmyer–Meshkov instability growth at a heavy–light interface via successive shocks is theoretically analysed and experimentally realized in a specific shock-tube facility. An analytical model is developed to forecast the interface evolution before and after the second shock impact, and the possibilities for the amplitude evolution pattern are systematically discussed. Based on the model, the parameter conditions for each scenario are designed, and all possibilities are experimentally realized by altering the time interval between two shock impacts. These findings may enhance the understanding of how successive shocks influence hydrodynamic instabilities in practical applications.

Mangroves are a natural defence of the coastal strip against extreme waves. Furthermore, innovative techniques of naturally based coast defence are used increasingly, according to the canons of eco-hydraulics. Therefore, it is important to correctly evaluate the transmission of waves through cylinder arrays. In the present paper, the attenuation of solitary waves propagating through an array of rigid emergent and submerged cylindrical stems on a horizontal bottom is investigated theoretically, numerically and experimentally. The results of the theoretical model are compared with the numerical simulations obtained with the smoothed particle hydrodynamics meshless Lagrangian numerical code and with experimental laboratory data. In the latter case, solitary waves were tested on a background current, in order to reproduce more realistic sea conditions, since the absence of circulation currents is very rare in the sea. The comparison confirmed the validity of the theoretical model, allowing its use for the purposes indicated above. Furthermore, the present study allowed for an evaluation of the bulk drag coefficient of the rigid stem arrays used, as a function of their density, the stem diameter, and their submergence ratio.