Our study concerns the propagation of acoustic waves through a thin screen made of a periodic arrangement of air bubbles in water. The bubbles are oscillators of the Minnaert type whose dynamics is modified by the containment. This nonlinear dynamics is obtained in the time domain using asymptotic analysis and a homogenization technique involving three scales, those being the scale of a bubble, that of the array and eventually that of the wavelength. The resulting effective model is set in the water (the screen has disappeared) and it encapsulates the effect of the screen in a jump of the normal acoustic velocity. The jump is linked to the continuous version of the bubble radius which satisfies an equation of the Rayleigh–Plesset type. This allows us to highlight two important effects. Firstly, a bubble within the array has a much larger radiative damping than an isolated bubble. Secondly it perceives a pressure which differs from the acoustic pressure imposed by the source due to bubble–bubble interactions; it results in a term of mass correction deduced from the Green's function for a Laplace problem which accounts for the bubble arrangement. Our findings are exemplified by numerical experiments of the scattering of a short pulse in the linear and nonlinear regimes.

Heterogeneous nucleation is the most effective mechanism for the inception of phase transformation. Solid walls and impurities act as a catalyst for the formation of a new thermodynamic phase by reducing the activation energy required for a phase change, hence enhancing nucleation. The formation of vapour bubbles close to solid, ideally flat, walls is addressed here by exploiting a mesoscale description that couples diffuse interface modelling of the two-phase vapour–liquid system with fluctuating hydrodynamics, extending previous work by the authors on homogeneous nucleation. The technical focus of this work is to directly account for hydrophobic or hydrophilic walls through appropriate boundary conditions compliant with the fluctuation–dissipation balance, a crucial point in the context of fluctuating hydrodynamics theory. This methodology provides access to the complete dynamics of the nucleation process, from the inception of multiple bubbles up to their long-time macroscopic expansion, on time and spatial scales unaffordable by standard techniques for nucleation, such as molecular dynamics. The analysis mainly focuses on the effect of wall wettability on the nucleation rate, and, albeit qualitatively in agreement with classical nucleation theory predictions, it reveals several discrepancies to be ascribed to layering effects in the liquid close to the boundary and to bubble–bubble interactions. In particular, it is found that, close to moderately hydrophilic surfaces, the most probable nucleation events occur away from the wall through a homogeneous mechanism.

We propose an automatable data-driven methodology for robust nonlinear reduced-order modelling from time-resolved snapshot data. In the kinematical coarse-graining, the snapshots are clustered into a few centroids representing the whole ensemble. The dynamics is conceptualized as a directed network, where the centroids represent nodes and the directed edges denote possible finite-time transitions. The transition probabilities and times are inferred from the snapshot data. The resulting cluster-based network model constitutes a deterministic–stochastic grey-box model resolving the coherent-structure evolution. This model is motivated by limit-cycle dynamics, illustrated for the chaotic Lorenz attractor and successfully demonstrated for the laminar two-dimensional mixing layer featuring Kelvin–Helmholtz vortices and vortex pairing, and for an actuated turbulent boundary layer with complex dynamics. Cluster-based network modelling opens a promising new avenue with unique advantages over other model-order reductions based on clustering or proper orthogonal decomposition.

We propose an image-based flow decomposition developed from the two-dimensional (2-D) tensor empirical wavelet transform (EWT) (Gilles, IEEE Trans. Signal Process., vol. 61, 2013, pp. 3999–4010). The idea is to decompose the instantaneous flow data, or their visualisation, adaptively according to the averaged Fourier supports for the identification of spatially localised structures. The resulting EWT modes stand for the decomposed flows, and each accounts for part of the spectrum, illustrating fluid physics with different scales superimposed in the original flow. With the proposed method, decomposition of an instantaneous three-dimensional (3-D) flow becomes feasible without resorting to its time series. Examples first focus on the interaction between a jet plume and 2-D wake, where only experimental visualisations are available. The proposed method is capable of separating the jet/wake flows and their instabilities. Then the decomposition is applied to an early-stage boundary layer transition, where direct numerical simulations provided a full dataset. The tested inputs are the 3-D flow data and their visualisation using streamwise velocity and $\lambda _{2}$ vortex identification criterion. With both types of inputs, EWT modes robustly extract the streamwise-elongated streaks, multiple secondary instabilities and helical vortex filaments. Results from 2-D stability analysis justify the EWT modes that represent the streak instabilities. In contrast to proper orthogonal decomposition or dynamic modal decomposition that extract spatial modes according to energy or frequency, EWT provides a new strategy for decomposing an instantaneous flow from its spatial scales.

A slowly varying modes solution of Wentzel–Kramers–Brillouin type is derived for the problem of sound propagation in a slowly varying two-dimensional duct with homentropic inviscid sheared mean flow and acoustically lined walls of slowly varying impedance. The modal shape function and axial wavenumber are described by the Pridmore-Brown eigenvalue equation. The slowly varying modal amplitude is determined in the usual way by an equation resulting from a solvability condition. For a general mean flow, this equation can be solved in the form of an incomplete adiabatic invariant. Due to conservation of specific mean vorticity along streamlines, two simplifications prove possible for a linearly sheared mean flow: (i) an analytically exact approximation for the mean flow, and (ii) a complete adiabatic invariant for the acoustics. For this last configuration some example cases are evaluated numerically, where the Pridmore-Brown eigenvalue problem is solved by a Galerkin projection combined with an efficient nonlinear iteration.

A nonlinear Cauchy–Poisson problem with impulsive surface forcing is investigated analytically and numerically. An incompressible liquid with an initially horizontal surface is instantaneously put into motion by an impulsive surface pressure distribution turned on and off during an infinitesimal time interval. We consider symmetric, antisymmetric and asymmetric pressure impulses based on dipoles and quadrupoles. The subsequent inviscid free-surface flow is governed by fully nonlinear surface conditions, which are solved exactly to third order in a small-time expansion. The small-time expansion applies to flows dominated by inertia. Such flows are generated by relatively strong pressure impulses, measured in gravitational units. We solve the problem numerically and find that only relatively weak pressure impulses will lead to oscillatory waves. The free surface will break before a full gravitational oscillation is completed when the amplitude of the pressure impulse exceeds one gravitational unit.

A fully nonlinear Cauchy–Poisson problem is investigated analytically by a small-time expansion. The inviscid incompressible fluid layer has an initially horizontal surface. The fluid is forced into motion by an impulsive surface pressure. The early nonlinear free-surface problem is solved to second order in a small-time expansion by the Lagrangian description of motion. Comparisons are made with two other solution procedures for the same nonlinear problem in the absence of gravity: a third-order small-time expansion and a numerical solution, based on full nonlinearity according to the standard Eulerian description. Good agreement is found between the present second-order Lagrangian solution and the previous third-order Eulerian solution, until both these asymptotic expansions diverge rather abruptly at the same time.

We investigate the large-scale circulation (LSC) of turbulent Rayleigh–Bénard convection in a large box of aspect ratio $\varGamma =32$ for Rayleigh numbers up to $Ra=10^9$ and at a fixed Prandtl number $Pr=1$. A conditional averaging technique allows us to extract statistics of the LSC even though the number and the orientation of the structures vary throughout the domain. We find that various properties of the LSC obtained here, such as the wall-shear stress distribution, the boundary layer thicknesses and the wind Reynolds number, do not differ significantly from results in confined domains ($\varGamma \approx 1$). This is remarkable given that the size of the structures (as measured by the width of a single convection roll) more than doubles at the highest $Ra$ as the confinement is removed. An extrapolation towards the critical shear Reynolds number of $Re_s^{{crit}} \approx 420$, at which the boundary layer (BL) typically becomes turbulent, predicts that the transition to the ultimate regime is expected at $Ra_{{crit}} \approx {O}(10^{15})$ in unconfined geometries. This result is in line with the Göttingen experimental observations (He et al., Phys. Rev. Lett., vol. 108, 2012, 024502; New J. Phys., vol. 17, 2015, 063028). Furthermore, we confirm that the local heat transport close to the wall is highest in the plume impacting region, where the thermal BL is thinnest, and lowest in the plume emitting region, where the thermal BL is thickest. This trend, however, weakens with increasing $Ra$.

A cylindrical and inclined jet in crossflow is studied under two distinct velocity ratios, $r=1$ and $r=2$, using highly resolved large eddy simulations. First, an investigation of turbulent scalar mixing sheds light onto the previously observed but unexplained phenomenon of negative turbulent diffusivity. We identify two distinct types of counter gradient transport, prevalent in different regions: the first, throughout the windward shear layer, is caused by cross-gradient transport; the second, close to the wall right after injection, is caused by non-local effects. Then, we propose a deep learning approach for modelling the turbulent scalar flux by adapting the tensor basis neural network previously developed to model Reynolds stresses (Ling et al., J. Fluid Mech., vol. 807, 2016a, pp. 155–166). This approach uses a deep neural network with embedded coordinate frame invariance to predict a tensorial turbulent diffusivity that is not explicitly available in the high-fidelity data used for training. After ensuring analytically that the matrix diffusivity leads to a stable solution for the advection diffusion equation, we apply this approach in the inclined jets in crossflow under study. The results show significant improvement compared to a simple model, particularly where cross-gradient effects play an important role in turbulent mixing. The model proposed herein is not limited to jets in crossflow; it can be used in any turbulent flow where the Reynolds averaged transport of a scalar is considered.

Numerical simulations are employed to investigate the dynamical separation of an initially stationary sphere from the surface of a two-dimensional ramp in hypersonic flow. We consider the inviscid limit, which is equivalent to assuming the sphere radius to be much larger than the ramp boundary-layer thickness; this assumption allows a range of Mach numbers and ramp angles to be explored efficiently. Of particular interest is determining how the shock-surfing phenomenon discovered by Laurence & Deiterding (J. Fluid Mech., vol. 676, 2011, pp. 396–431), in which a spherical body can stably oscillate about an oblique shock as it moves downstream, manifests itself in such a situation. First, the isolated interactions between a sphere and an oblique shock, and then between a sphere and an inviscid wall, are examined independently to elucidate relevant trends. Full trajectory predictions are subsequently performed using a decoupled model in which the shock and wall interactions are assumed to contribute independently to the aerodynamic forces. Three types of trajectories are found to be possible: surfing of the spherical body; initial expulsion outside the shock layer followed by re-entry and entrainment; or direct entrainment. At relatively low hypersonic Mach numbers, the latter two types of trajectories are predominant, but at higher Mach numbers ($M\gtrsim 10$), surfing becomes possible over an increasingly wide range of ramp angles and downstream release locations. By reparameterizing the release location as the initial lateral distance of the sphere from the shock, good collapse of the transition boundary delineating surfing from ejection/re-entrainment over various Mach numbers and ramp angles is obtained.

The separation dynamics of a sphere released from the surface of a ramp into a hypersonic flow is investigated, focusing on the influence of the ramp boundary layer on the sphere behaviour. First, numerical simulations are conducted of a sphere interacting with an isolated high-speed boundary layer to determine the influence on the sphere force coefficients as the sphere diameter and wall-normal location are varied. It is found that the lift coefficient is strongly affected by the near-wall interactions, becoming increasingly negative as the ratio of the sphere radius to boundary-layer thickness, $r/\delta$, is decreased. These results are combined with force coefficients derived from simulations of the sphere interacting with the ramp-generated oblique shock to enable numerical predictions of the sphere trajectories for a $10^{\circ }$ ramp at Mach 6 (using a similar decoupled approach to Part 1 of this work). It is found that the three trajectory types of the inviscid situation – shock surfing, ejection followed by re-entrainment within the shock layer and direct entrainment – also characterize the sphere behaviour here. Their relative prevalence, however, is influenced by the sphere size: for smaller values of $r/\delta$, direct entrainment dominates because of the wall suction, while shock surfing and then ejection/re-entrainment become increasingly likely at larger values of $r/\delta$. Increasing the ramp angle and/or the free-stream Mach number reduces the relative influence of the boundary-layer interactions. Finally, experiments are conducted using free-flying spheres released from a ramp surface in a hypersonic shock tunnel, confirming the major trends predicted numerically.