Building on the hypothesis of wall and jet structural modes proposed in Part 1 (Choudhary et al. 2024), this study reports coherent patterns and vortical structures associated with the jet mode by further analysing our experimental particle image velocimetry datasets. Instantaneous velocity fields are binned based on dominant streamwise Fourier modes, focusing on submodes with wavelengths $\lambda _x\approx 5{z_{T}}$ (submode 1) and $\lambda _x\approx 2.5{z_{T}}$ (submode 2); $z_{T}$ is outer length scale of the flow. Two-point correlations of streamwise velocity fluctuations for the total and modal fields reveal near-periodic coherent patterns inclined backwards (∼$14^{\circ }$) in the outer region and forwards (∼$9^{\circ }$) in the inner region. Vortical structures in conditionally averaged velocity fluctuation vector fields are examined using linear stochastic estimation (LSE) with anticlockwise vorticity (prograde) at the outer energy site as the condition. The vortical structure of submode 1 is a three-vortex system with (i) a robust clockwise vortex in the inner region and (ii) a saddle-point topology in the outer region. The vortical structure of submode 2 is a backward-leaning vortex packet. The LSE fields indicate Q1–Q3 events in the inner region contributed by ‘non-local’ eddies through the interaction of outer and inner submode 1 vortices. Quadrant analysis reveals that Q1–Q3 events due to ‘non-local’ eddies outweigh Q2–Q4 contributions of ‘local’ eddies, producing counter-gradient momentum diffusion below mean velocity maximum. These findings further substantiate the hypothesis of wall and jet structural modes and indicate that the region below mean velocity maximum in wall jets significantly differs from a turbulent boundary layer.

The wake systems of a conventional ducted propeller ($\mathrm{DP}$) and a rim driven thruster ($\mathrm{RDT}$) are compared. The latter is an innovative ducted propeller, whose blades are installed on a rim rotating within the nozzle, with their tips oriented inwards and no need of a rotating hub. The flow was reproduced by large eddy simulation (LES) on a cylindrical grid consisting of 6.3 billion points. Substantial deviations between the flow physics downstream of the two propellers are revealed and an order of magnitude drop in the pressure minima and turbulent stresses is found across the rotor and in the wake of $\mathrm{RDT}$. These changes are mainly attributable to the absence of the hub vortex, the helical vortices from the root of the blades, and the leakage flow generated between their tip and the inner surface of the nozzle. In the rim driven thruster, they are replaced by inner, helical vortices shed from the tip of its blades. In addition, the trailing wake of the blades of $\mathrm{RDT}$ is populated by smaller streamwise vortices and lower turbulence levels. This is due to their modified design, characterised by a more uniform spanwise distribution of the load, allowed by the absence of both the hub and the leakage between the tip of the blades and the nozzle. In conventional ducted propellers, they require a reduction of the load of the blades towards their root and their tip with the purpose of mitigating the intensity of the hub vortex and the leakage flow, respectively.

In this study, we investigate the modulation of sustained turbulence by settling spherical particles using particle-resolved direct numerical simulations. Gravity effects are studied by varying the Galileo number, ${\textit{Ga}}$, through the particle–fluid density ratio for particles of Taylor-scale size. Particle sedimentation causes enhanced viscous dissipation and anisotropy of the fluid velocity fluctuations, which increase with ${\textit{Ga}}$. More significantly, the energy spectra exhibit a $\kappa ^{-3}$ scaling, which coexists with the classic $\kappa ^{-5/3}$ law for particle-laden turbulence with weak sedimentation (${\textit{Ga}} \lesssim 40$); the $\kappa ^{-3}$ scaling widens its wavenumber range with ${\textit{Ga}}$ and dominates the energy spectrum at the highest ${\textit{Ga}}$ under study. The scale-by-scale energy budgets demonstrate that the particle–fluid interactions mainly transfer energy from large to small scales for small settling speeds, whereas the particle sedimentation injects energy into the carrier flow and breaks the isotropic energy distribution for a high ${\textit{Ga}}$. In particular, the gravity-induced forcing originates vertically elongated structures while disrupting horizontal flow correlations, thereby altering the nonlinear interscale energy transfer and reshaping the energy spectra. For the solid phase, it is found that the particle vertical velocity fluctuations decrease with ${\textit{Ga}}$, and turbulence has a weaker influence on the mean settling velocity for particles with a higher ${\textit{Ga}}$. Moderate particle clustering is identified, an observation not changing significantly with the particle settling speed. The collision rate among particles is highest at the strongest particle sedimentation, due to the enhanced relative radial velocity of particle pairs which are preferentially horizontally aligned.

We investigate droplet deformation following laser-pulse impact at low Weber numbers (${\textit{We}}\sim 0.1{-}100$). Droplet dynamics can be characterised by two key parameters: the impact Weber number and the width, W, of the distribution of the impact force over the droplet surface. By varying laser-pulse energy, our experiments traverse a phase space comprising (i) droplet oscillation, (ii) breakup or (iii) sheet formation. Numerical simulations complement the experiments by determining the pressure width and by allowing We and W to be varied independently, despite their correlation in the experiments. A single phase diagram, integrating observations from both experiments and simulations, demonstrates that all phenomena can be explained by a single parameter: the deformation Weber number ${\textit{We}}_{{d}}=f({\textit{We}},{W})$ that is based on the initial radial expansion speed of the droplet, following impact. The resulting phase diagram separates (i) droplet oscillation for ${\textit{We}}_{{d}}\lt 5$, from (ii) breakup for $5\lt {\textit{We}}_{{d}}\lt 60$ and (iii) sheet formation for ${\textit{We}}_{{d}}\gt 60$.

The interaction of a spark-generated cavitation bubble with an initially perturbed free surface is investigated experimentally, numerically and analytically. By exploiting contact-line pinning, we accurately prescribe an initial meniscus with a thin, hydrophilic-coated rod inserted into the liquid. A pronounced surface cavity, driven by the oscillating bubble, forms and penetrates downward to a scale comparable to the bubble itself. The coupled cavity–bubble system exhibits two distinct regimes – coalescence and non-coalescence – separated by a critical condition governed by the non-dimensional stand-off parameter $\gamma$ and the initial meniscus height $h_m$. In the non-coalescence regime, the cavity evolves through inception, expansion and rebound/jetting. The maximum cavity length $h_{\textit{c}}$ follows a power-law scaling $h_{\textit{c}}\propto \gamma ^{\alpha }$ with $\alpha =-2.7$ (experiments) and $\alpha =-2.6$ (simulations) for $1.5\lesssim \gamma \lesssim 3$, where inertia dominates. Deviations emerge for $\gamma \lesssim 1.5$ (strong nonlinearity) and $\gamma \gtrsim 3$ (surface tension and viscosity become noticeable). An analytical model based on the Rayleigh–Plesset equation combined with nonlinear Rayleigh–Taylor instability theory captures the trend and confirms that $h_m$ plays only a secondary role relative to $\gamma$. In the coalescence regime, atmospheric air vents into the bubble through the merged cavity, weakening the collapse intensity and reducing the associated pressure peak. We also examine air/liquid compressibility and boundary layer effects, whose significance grows as $\gamma$ decreases. These findings are relevant to surface-jetting technologies, cavitation-erosion mitigation and underwater-noise suppression.

Numerical solutions of the Navier–Stokes equations in the problem of subsonic jets obtained using a scheme with the 24th-order multi-operators-based dissipation-free approximations are presented and analysed. The spectral properties of the scheme obtained as a result of optimisation for high resolution of small solutions scales are described. The role of added multi-operators in excitation of instability, which make a 23rd-order contribution to the approximation error and form a dissipative component of the scheme, is studied. It is shown how well-known structures of jets with laminar regions of instability development, coherent structures and turbulent regimes at low and moderately high Reynolds numbers are displayed in numerical solutions. Comparisons with experimental data and results of calculations based on large-eddy simulations are given. The results of statistical processing with the calculation of autocorrelations are presented. In particular, their changes with distance from the nozzle are given, characterising different flow regimes at the considered Reynolds numbers. Comparison of spatial and temporal autocorrelations in the region of the turbulent regime indicate good fulfilment of Taylor’s hypothesis on frozen turbulence.

Turbulent pipe flow is of substantial importance in practical applications, and it remains challenging to depict the characteristic complex multiscale dynamics by a unified theoretical framework, hindered by its inherent intermittency. Inspired by a recent study of velocity circulation in turbulent channel flows from Duan, Chen & Sreenivasan (2025 J. Fluid Mech., vol. 1009, p. R4), in this study, we investigate the statistical characteristics of velocity circulation (or equally the area integral of wall-normal vorticity) over rectangular loops in concentric cylindrical shells, parallel to the pipe wall. The statistics are implemented using direct numerical simulation data at friction Reynolds numbers of $ \textit{Re}_\tau =1057$ and $2000$. Close to the pipe wall, the circulation in the inertial range resides on space-filling unifractal sets, with the Hölder exponent smaller than Kolmogorov’s $4/3$. Away from the pipe wall, the circulation displays bifractal characteristics and the Hölder exponents for high moment orders are very close to those reported in channel flows and homogeneous isotropic turbulence. The circulation statistics are only dependent on the area enclosed by the loops, and are invariant to the loop aspect ratio, once both edge lengths of the loops are in the inertial range.

Traditional Reynolds-averaged Navier–Stokes (RANS) closures, based on the Boussinesq eddy-viscosity hypothesis and calibrated on canonical flows, often yield inaccurate predictions of both mean flow and turbulence statistics. Here, we consider flow past a circular cylinder over a range of Reynolds numbers ($3900$–$100\,000$) and Mach numbers ($0$–$0.3$), encompassing incompressible and weakly compressible regimes, with the goal of improving predictions of mean velocity and Reynolds forces. To this end, we assemble a cross-validated dataset comprising hydrodynamic particle image velocimetry (PIV) in a towing tank, aerodynamic PIV in a wind tunnel and high-fidelity spectral element direct numerical simulation and large eddy simulation. Analysis of these data reveals a universal distribution of Reynolds stresses across the parameter space, which provides the foundation for a data-driven closure. We employ physics-informed neural networks (PINNs), trained with the unclosed RANS equations, to infer the velocity field and Reynolds-stress forcing from boundary information alone. The resulting closure, embedded in a forward PINN solver and the numerical solver OpenFOAM, significantly improves RANS predictions of both mean flow and turbulence statistics relative to conventional models.

Electrohydrodynamic (EHD) instabilities at polymer–porous interfaces play a pivotal role in determining interfacial morphology, wettability and pattern formation, with implications for energy storage, diagnostics and flexible electronics. This study presents a comprehensive general linear stability analysis to examine electric-field-induced instabilities at a confined interface between a viscoelastic polymer gel and a saturated porous medium. By coupling Maxwell stresses with a modified Darcy–Brinkman–Kelvin–Voigt framework, the model captures how porous medium-moderated EHD instabilities influence both the onset and dominant instability modes. Key parameters – including the electric Rayleigh number, Darcy number, dielectric contrast and geometric filling ratio – govern the spatio-temporal features of emerging patterns. The analysis reveals a sigmoidal dependence of characteristic length and time scales on permeability, i.e. Darcy number, establishing three regimes: impermeable, transitional and highly permeable, with a shift toward shorter wavelengths. The length and time scale transitions, triggered by the solid-saturated porous medium, are further moderated by the dielectric contrast – instabilities are suppressed when the contrast is low and amplified when it is high, enabling sub-micron patterning. Geometric confinement, i.e. increasing filling ratio, further intensifies pattern length scales, suggesting the feasibility of fabricating complex ultra-fine nanoscale encapsulated porous patterns. The elasticity of the viscoelastic layer imposes a threshold for instability onset and is critical for identifying wettability transitions at the interface. This framework offers predictive insight into tuning instability modes through permeability–viscoelasticity–electrostatics interplay, laying the foundation for wettability-controlled interfaces and self-organised interfacial patterns in next-generation EHD-driven systems.