In the present study, we propose a novel skin-friction prediction formula based on a re-established self-similarity within the adverse-pressure-gradient (APG) turbulent boundary layer. The basic idea lies in introducing a novel velocity scale, which is derived mathematically and adapted physically from the linear total stress within the boundary layer. This scale assimilates concurrently and fundamentally the friction velocity, two distinct pressure-gradient velocity scales and the half-power law of the mean velocity in the intermediate region. Then, this scale formula is well validated across a comprehensive, multi-geometry database of APG flows over flat plates, curved plates, ramps and airfoils, which covers an unprecedented parameter range, with friction Reynolds number ranging from $10^2$ to $5\times 10^3$ and the Rotta–Clauser pressure-gradient parameter spanning from $10^{-1}$ to $10^2$. Crucially, the proposed scale consistently recovers a classical logarithmic region across all tested APG conditions, thereby restoring the self-similar structures traditionally absent in strong or non-equilibrium pressure-gradient flows. Leveraging this reconstructed self-similarity, we further formulate a new, robust skin-friction prediction model which demonstrates predictive errors confined within $\pm 20\,\%$ for all the investigated non-equilibrium flow states.

The evolution of main shock waves generated by multiple finite-energy blast sources with time-delayed energy release is investigated. We demonstrate that a merged shock wave forms when the individual blasts interact within a critical interval of delay times. This interval is determined principally by the initial sound speed ratio between the compressed gas of the blasts and the ambient medium. Numerical simulations confirm both the existence and the boundaries of this merging regime. Extending the analysis to the strong-shock stage, we derive an asymptotic solution that incorporates an equivalent energy release and a temporal scaling for the dual-source system, thereby describing the propagation of the merged shock front. These results provide a foundational model for multiple blast–shock processes, with potential applications to astrophysical phenomena, intense laser–matter interactions, and blast-wave dynamics in general.

An asymptotic theory is developed to investigate the interaction of surface water waves and a compressible muddy seabed in water of intermediate depth. The water column is treated as inviscid outside a thin bottom boundary layer, the thickness of which is assumed to be of the same order of magnitude as the wave amplitude. The seabed is modelled as an isotropic and homogeneous poro-viscoelastic layer with an incompressible solid skeleton and a compressible pore fluid. The thickness of the seabed is assumed to be comparable to the wave amplitude. Using a perturbation approach, the leading-order analytical solutions for the wave and mud flow are derived, while the evolution of the wave envelope, the mean Eulerian velocity and the mass transport velocity beneath progressive waves are obtained at the second order. Based on the present solution, the wave motion and the induced mass transport are analysed and compared with previous solutions that ignore the compressibility of the seabed. The results demonstrate that the compressibility of the seabed plays a critical role in modulating the flows within both the seabed and the overlying bottom boundary layer. Neglecting compressibility may lead to an underestimation of the interface vertical displacement in highly elastic beds and an overestimation in viscous-dominated cases. Consequently, the Reynolds stress distributions in these regions deviate significantly from predictions based on incompressible seabed theory. This inaccuracy further propagates to the prediction of second-order steady currents and mass transport velocities.

We investigate the onset of transient natural convection in fluid layers subject to volumetric radiative heating and surface cooling. Linear stability analysis reveals a non-monotonic evolution of stability in deep layers, where the flow undergoes successive stages of initial destabilisation, intermediate suppression and eventual restabilisation. This complex temporal behaviour necessitates the definition of dual critical Rayleigh numbers: a lower bound marking the onset of initial instability and an upper bound required for sustained convection. To efficiently predict these thresholds, we develop a local Rayleigh number model that depends solely on the instantaneous conductive temperature profile. When the Rayleigh number exceeds the lower bound, the critical time $t_c$ for flow onset is determined through transient linear stability analysis, and scaling laws are derived to characterise the dependence of $t_c$ on the Rayleigh number $\textit{Ra}$, Prandtl number $\textit{Pr}$, cooling parameter $\phi$ and layer depth $H$. Two distinct instability triggering mechanisms are identified: a top-triggered regime, where $t_c \sim [\textit{Ra} \textit{Pr} \phi /(2+\textit{Pr})]^{-1/2}$, and a bottom-triggered regime, where $t_c \sim [\textit{Ra} \textit{Pr} \exp (-H)/(2+\textit{Pr})]^{-1/2}$. All theoretical predictions are rigorously validated against direct numerical simulations, providing a unified predictive framework for convective onset in systems governed by coupled effects of radiation absorption and surface cooling.

This study investigates the influence of a rear-attached splitter plate on the vortex dynamics and turbulence characteristics in the wake of a circular cylinder. Three-dimensional direct numerical simulations (DNS) are performed at a turbulent Reynolds number of 1000 and non-dimensional plate lengths ($ L/D$) of 0–4. The splitter plate affects the vortex dynamics and turbulence characteristics in a highly non-monotonic manner. Specifically, both the strength of the primary vortices and the turbulent kinetic energy in the wake decrease with increasing $ L/D$ over $ L/D$ = 0–1.5, followed by a local increase over $ L/D$ = 1.5–2 and another decrease over $ L/D$ ≥ 2. The abnormal local increase is because the vortex formation location transitions from downstream of the splitter plate for $ L/D$ ≤ 1.5 to the two sides of the plate for $ L/D$ ≥ 2. Owing to the presence of the primary vortex street in the wake, the turbulence in the wake is strongly anisotropic both globally and locally. After removing the contribution from the coherent primary vortices, the wake becomes much more isotropic globally and homogeneous locally. The present DNS dataset also enables an evaluation of several widely used surrogate models for the kinetic energy dissipation rate. A major finding is that a minor increase in the contribution of the coherent component (e.g. for $ L/D$ ≥ 2) may strongly deteriorate the applicability of the surrogate models of local axisymmetry and local homogeneity. In general, this study provides new insights and mechanisms for the control of turbulence in bluff-body wakes.

The instability of baroclinic Rossby waves over two-dimensional topography is examined using a nonlinear model, a linear stability calculation and wave triads. In all cases, the Rossby waves are unstable, as seen previously over a flat bottom. But topography decreases the growth rates and changes the structure of the unstable waves. When the topographic height (or slope) exceeds a critical value, the instability is ‘locked’ to topography, in that the most unstable mode, particularly in the lower layer, resembles the bathymetry. In this limit, the growth rate becomes independent of topographic height. A triad calculation suggests that the growth rates in the locked state should depend on the lateral scale of the bathymetry but not its height, and that locking does not occur for topographic scales smaller than the surface deformation radius. The results suggest an alternate way that topographically locked flow can be generated, and indicate that baroclinic instability can be much different over steep bathymetry.

Interfacial instability dominates the dynamics as a cavitation bubble oscillates in close proximity to a liquid surface, driving perturbations on both the bubble wall and the liquid surface. The penetration of the liquid layer initiates ventilation, exposing the bubble interior and thereby altering its subsequent dynamics. To quantitatively elucidate the interfacial coupling-induced instability, we develop a theoretical model that couples the perturbation equation with the bubble oscillation equation, considering the liquid viscosity. The model predicts the transition boundaries between ventilation patterns by critical stand-off parameters, which scale exponentially with the liquid viscosity to the −1/3 power. The boundary between complete and partial ventilation regimes shows negligible viscous dependence due to the vanishingly short perturbation growth time. Furthermore, we derive the scaling law of ventilation time, defining it as the instant of perturbation penetration. A series of experiments on bubble oscillation near a liquid surface was conducted, which verified the predictions of the theoretical model. This offers a practical framework for the engineering application of near-surface bubble collapse.

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.