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.

This article presents a first-principles model for electrosprays operating in the ion emission regime. The model considers ion emission from a Taylor cone anchored on a tubular emitter, including the fluid dynamics as well as the electrostatic interaction between the liquid, the electrodes and the ion beam. The model accounts for the self-heating of the liquid due to ohmic and viscous dissipation, and the associated variation of the viscosity and electrical conductivity with temperature. The numerical solution reproduces the experimental phenomenology of the ion emission regime (e.g. current levels, the high sensitivity to the ion solvation energy, the proportionality between the emitted current and emitter potential, etc.), and other aspects of the underlying physics such as the coupling between ion emission and self-heating of the liquid. The numerical solution is also used to validate a simpler analytical model that yields scaling laws for the emitted current, and for the characteristic length, current density and electric field of the emission region. The analytical model also provides liquid-dependent criteria for the onset of the ion emission regime.

The description of weakly nonlinear water-wave evolution over a horizontal bottom by the integro-differential Zakharov equation, because of utilising the underlying Hamiltonian structure, has many advantages over direct use of the Euler equations. However, its application to finite-depth situations is not straightforward since, in contrast to the deep-water case, the kernels governing the four-wave interactions are singular, as well as the kernels in the canonical transformation that removes non-resonant interactions from the original equations of motion. At the singularities, these kernels are finite but not unique. The issue of how to use the Zakharov equation for finite depth and whether it is possible at all was debated intensely in the literature for decades but remains outstanding. Here we show that the absence of a limit of the kernels at the singularities is inconsequential, since in the equations of motion it is only the integral that matters. By applying the definition of the Dirac-$\delta$, we show that all the integrals involving a trivial manifold singularity are evaluated uniquely. Therefore, the Zakharov evolution equation and the nonlinear canonical transformation are only apparently singular. The findings are validated by application to examples where predictions based on the Zakharov equation are compared with known solutions obtained from the Euler equations.

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.

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.

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.

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.

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.

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.