The interaction between near-wall turbulence and wall curvature is described for the incompressible flow in a plane channel with a small concave–convex–concave bump on the bottom wall, with height comparable to the wall-normal location of the main turbulent structures. The analysis starts from a database generated by a direct numerical simulation and hinges upon the anisotropic generalised Kolmogorov equations, i.e. the exact budget equations for the second-order structure function tensor. The influence of the bump on the wall cycle and on the energy production, redistribution and transfers is described in the physical and scale spaces. Over the upstream side of the bump, the energy drained from the mean flow to sustain the streamwise fluctuations decreases, and the streaks of high and low streamwise velocity weaken and are stretched spanwise. After the bump tip, instead, the production of streamwise fluctuations grows and the streaks intensify, progressively recovering their characteristic spanwise scale. The wall-normal fluctuations, and thus the quasi-streamwise vortices, are sustained by the mean flow over the upstream side of the bump, while energy flows from the vertical fluctuations to the mean field over the downstream side. On the concave portion of the upstream side, the near-wall fluctuations form structures of spanwise velocity which are consistent with Taylor–Görtler vortices at an early stage of development. Their evolution is described by analysing the scale-space pressure–strain term. A schematic description of the bump flow is presented, in which various regions are identified according to the signs of curvature and streamwise pressure gradient.

We present results for Newtonian laminar miscible displacement flows in a narrow, vertical, eccentric annulus, obtained from experiments carried out in a scaled laboratory set-up. These are compared with two computational models introduced in Part 1, Zhang & Frigaard (J. Fluid Mech., vol. 947, 2022, A732). Quite close matches have been found for different approaches, regarding the overall evolving displacement process: front shape, dispersion levels and front velocities. Standardized criteria have been established to identify whether a flow has steady/unsteady and dispersive/non-dispersive displacement front, applicable to both concentric and eccentric annuli. Three characteristic flows have been observed, unsteady and dispersive, dispersive with steady front, and non-dispersive with steady front. Through both qualitative and quantitative studies, the importance of the buoyancy number has been established: it both restrains dispersion on the annular gap scale and induces secondary flows in the azimuthal direction.

Bathymetry is an important factor affecting wave propagation in coastal environments but is often challenging to measure in practice. We propose a method for inferring coastal bathymetry from spatial variations in surface waves by combining a high-order spectral method for wave simulation and an adjoint-based variational data assimilation method. Recursion-formed adjoint equations are derived to obtain the sensitivity of the wave surface elevation to the underlying bottom topography to any desired order of nonlinear perturbation. We also develop a multiscale optimisation method to eliminate spurious high-wavenumber fluctuations in the reconstructed bathymetry data caused by sensitivity variations over the different length scales of surface waves. The proposed bottom detection method is validated with a realistic coastal wave environment involving complex two-dimensional bathymetry features, non-periodic incident waves and nonlinear broadband multidirectional waves. In numerical experiments at both laboratory and field scales, the bathymetry reconstructed from our method agrees well with the ground truth. We also show that our method is robust against imperfect surface wave data in the presence of limited sampling frequency and noise.

We investigate the statistics of turbulence in emulsions of two immiscible fluids of the same density. We compute velocity increments between points conditioned to be located in the same phase or in different phases, and examine their probability density functions (PDFs) and the associated structure functions (SFs). This enables us to demonstrate that the presence of the interface reduces the skewness of the PDF at small scales and therefore the magnitude of the energy flux towards the dissipative scales, which is quantified by the third-order SF. The analysis of the higher-order SFs shows that multiphase turbulence is more intermittent than single-phase turbulence. In particular, the local scaling exponents of the SFs display a saturation below the Kolmogorov–Hinze scale, which indicates the presence of large velocity gradients across the interface. Interestingly, the statistics of the velocity differences in the carrier phase recovers that of single-phase turbulence when the viscosity of the dispersed phase is high.

A mathematical and numerical framework is proposed to compute the displacement and merging dynamics of sliding droplets under the action of a constant shear exerted by a gas flow. An augmented formulation is implemented to model surface tension including the full curvature of the free surface. A set of shallow-water evolution equations is obtained for the film thickness, the averaged velocity, an additional quantity (with dimension of a velocity) taking into account the capillary effects and a tensor called enstrophy. The enstrophy accounts for the deviation of the velocity profile from a constant velocity distribution. The formulation is consistent with the long-wave expansion of the basic equations with a conservative part and source terms including the effect of viscosity, in the form of a viscous friction and the effect of the shear stress. The model is hyperbolic with generalised diffusion terms due to capillarity. Finally, our model is completed with a disjoining pressure formulation that is able to account for the hysteresis of the static contact angle. In this formulation, the advancing or receding nature of the contact line is assessed by the accumulation or reduction of mass of the droplet at the contact line. Simulations of sliding water droplets are performed with periodic boundary conditions in a domain of limited size. Hysteresis of the static contact angle causes a slowdown of the drops and a delay in the sequence of coalescence of the drops.

The large-eddy simulation technique was used to investigate the dynamics of unsteady flow separation on a flat-plate turbulent boundary layer. The unsteadiness was generated by imposing an oscillating, wall-normal velocity profile at the top of the computational domain, and a range of reduced frequencies ($k$), from a very rapid flutter-like motion to a slow quasi-steady oscillation, was studied. Ambrogi et al. (J. Fluid Mech., vol. 945, 2022, A10) showed that the reduced frequency greatly affects the transient separation process, and at a frequency $k=1$, the separation region became unstable and was advected periodically out of the domain. In this paper, we discuss the causes of the observed advection process and the effects of the unsteadiness on the second moments. The time evolution of turbulent kinetic energy, for instance, reveals that an advection-like phenomenon is also present at a very low reduced frequency, but its dynamic behaviour is completely different from that of the intermediate frequency ($k=1$). At the intermediate frequency the entire recirculation region is advected downstream, keeping its shape. The advected structure is rotational in nature, and moves at constant speed. In contrast, in the low-frequency case the advected fluid originates at the reattachment point, and the structure is shear-dominated. Particle pathlines reflect the fact that the flow at the low frequency is quasi-steady-state, but show peculiar differences at the intermediate frequency, in which the flow response to the freestream forcing depends on the particle positions in the wall-normal direction.

The present work is devoted to the analysis of drop impact on a deep liquid pool, focusing on the high-energy splashing regimes caused by large raindrops at high velocities. Such cases are characterized by short time scales and complex mechanisms, thus they have received very little attention until now. The BASILISK open-source solver is used to perform three-dimensional direct numerical simulations. The capabilities of octree adaptive mesh refinement techniques enable capturing of the small-scale features of the flow, while the volume of fluid approach combined with a balanced-force surface-tension calculation is applied to advect the volume fraction of the liquids and reconstruct the interfaces. The numerical results compare well with experimental visualizations: both the evolution of crown and cavity, the emanation of ligaments, the formation of bubble canopy and the growth of a downward-moving spiral jet that pierces through the cavity bottom, are correctly reproduced. Reliable quantitative agreements are also obtained regarding the time evolution of rim positions, cavity dimensions and droplet distributions through an observation window. Furthermore, simulation gives access to various aspects of the internal flows, which allows us to better explain the observed physical phenomena. Details of the early-time dynamics of bubble ring entrapment and splashing performance, the formation/collapse of bubble canopy and the spreading of drop liquid are discussed. The statistics of droplet size show the bimodal distribution in time, corroborating distinct primary mechanisms of droplet production at different stages.

It is theoretically known that an isotropic chemically active particle in an unbounded solution undergoes symmetry breaking when the intrinsic Péclet number ${{Pe}}$ exceeds a finite critical value (Michelin et al., Phys. Fluids, vol. 25, 2013, 061701). At that value, a transition takes place from a stationary state to spontaneous motion. In two dimensions, where no stationary state is possible in an unbounded domain, a linear stability analysis in a large bounded domain (Hu et al., Phys. Rev. Lett., vol. 123, 2019, 238004) reveals that the critical ${{Pe}}$ value slowly diminishes as the domain size increases. Motivated by these findings, we here consider an unbounded domain from the outset, addressing the two-dimensional problem of steady self-propulsion with a focus on the limit ${{Pe}}\ll 1$. This singular limit is handled using matched asymptotic expansions, conceptually decomposing the fluid domain into a particle-scale region, where the leading-order solute transport is diffusive, and a remote region, where diffusion and advection are comparable. The expansion parameter is provided by the product of ${{Pe}}$ and $U$, the unknown particle speed (normalised by the standard autophoretic scale). The problem is unconventional in that the scaling of $U$ with ${{Pe}}$ must be determined in the course of the perturbation analysis. The resulting approximation, $U=4\exp ({-2/{Pe}-\gamma _{E}-1})/{{Pe}}$ ($\gamma _{E}$ being the Euler–Mascheroni constant), is in remarkable agreement with the numerical predictions of Hu et al. in the common interval of validity.

We construct reduced-order models of aeroacoustic sources for single and twin subsonic jets ($M_j=0.9$, $Re=3600$), with the goal of accurately recovering the far-field sound over a wide band of frequencies $St=[0.07,1.0]$ and directivity angles $\phi = [30^{\circ },120^{\circ }]$ within a subdecibel level accuracy. These models are realized via combining spatio-temporally coherent spectral proper orthogonal decomposition (SPOD) modes extracted directly from Lighthill's stress tensor, itself calculated using large-eddy simulation (LES). We consider two sets of twin subsonic jets of diameter $D$ each, with spacings of $0.1D$ and $1D$, where the jets merge upstream and downstream of breakdown, respectively. The closely spaced twin jet decays the slowest due to reduced turbulent stresses which are, however, more broadband due to early merging. Such jets show strong shielding in the plane of jets, especially at shallow directivity angles where sound levels may drop below that of the single jet. The farther spaced twin jets have dynamics more akin to the constituent single jet with turbulent fluctuations peaking here at $St=0.34$, but showing very little shielding, with their overall sound pressure level (OASPL) mostly linked to the nature of extra flow structures created during merging. Three-dimensional, energy-ranked, coherent structures for twin jets exhibit rather poor low-rank behaviour, especially at the far-field spectral peak $St=0.14$. At $St \gtrsim 0.3$, the SPOD wavepackets show strong visual coherence, resembling Kelvin–Helmholtz instability modes upstream of breakdown, while at the lower frequencies, there is very little spatial coherence with wavepackets peaking downstream of breakdown. Although the leading SPOD modes radiate poorly, reduced-order models using a subset of them, up to $45$ SPOD modes per frequency, show a remarkable match (within $1$ dB) against the LES-predicted sound over $0.1 \lesssim St \lesssim 0.5$, at all angles investigated. At other frequencies, the closely spaced twin jet shows more error, due to its greater hierarchy of spatio-temporal structures, showing slower convergence at the shallower angles.

We present new numerical solutions for nonlinear standing water waves when the effects of both gravity and surface tension are considered. For small values of the surface tension parameter, solutions are shown to exhibit highly oscillatory capillary waves (parasitic ripples), which are both time- and space-periodic, and which lie on the surface of an underlying gravity-driven standing wave. Our numerical scheme combines a time-dependent conformal mapping together with a shooting method, for which the residual is minimised by Newton iteration. Previous numerical investigations typically clustered gridpoints near the wave crest, and thus lacked the fine detail across the domain required to capture this phenomenon of small-scale parasitic ripples. The amplitude of these ripples is shown to be exponentially small in the zero surface tension limit, and their behaviour is linked to (or explains) the generation of an elaborate bifurcation structure.

In the present study, we investigate the compressibility effects in supersonic and hypersonic turbulent boundary layers under the influence of wall disturbances by exploiting direct numerical simulation databases at Mach numbers up to 6. Such wall disturbances enforce extra Reynolds shear stress on the wall and induce mean streamline curvature in rough wall turbulence that leads to the intensification of turbulent motions in the outer region. The turbulent and fluctuating Mach numbers, the density and the velocity divergence fluctuation intensities suggest that the compressibility effects are enhanced by the increment of the free-stream Mach number and the implementation of the wall disturbances. The differences between the Reynolds and Favre average due to the density fluctuations constitute approximately $9\,\%$ of the mean velocity close to the wall and $30\,\%$ of the Reynolds stress near the edge of the boundary layer, indicating their non-negligibility in turbulent modelling strategies. The comparatively strong compressive events behaving as eddy shocklets are observed at the free-stream Mach number of $6$ only in the cases with wall disturbances. By further splitting the velocity into the solenoidal and dilatational components with the Helmholtz decomposition, we found that the dilatational motions are organized as travelling wave packets in the wall-parallel planes close to the wall and as forward inclined structures in the form of radiated waves in the vertical planes. Despite their increased magnitudes and higher portion in the Reynolds normal and shear stresses, the dilatational motions show no tendency of contributing significantly to the skin friction and the production of turbulent kinetic energy due to their mitigation by the cross-correlation between the solenoidal and dilatational velocity components.