An experimental study of the dynamics and droplet production in three mechanically generated plunging breaking waves is presented in this two-part paper. In the present paper (Part 2), in-line cinematic holography is used to measure the positions, diameters ($d\geq 100\ \mathrm {\mu }{\rm m}$), times and velocities of droplets generated by the three plunging breaking waves studied in Part 1 (Erinin et al., J. Fluid Mech., vol. 967, 2023, A35) as the droplets move up across a horizontal measurement plane located just above the wave crests. It is found that there are four major mechanisms for droplet production: closure of the indentation between the top surface of the plunging jet and the splash that it creates, the bursting of large bubbles that were entrapped under the plunging jet at impact, splashing and bubble bursting in the turbulent zone on the front face of the wave and the bursting of small bubbles that reach the water surface at the crest of the non-breaking wave following the breaker. The droplet diameter distributions for the entire droplet set for each breaker are fitted with power-law functions in separate small- and large-diameter regions. The droplet diameter where these power-law functions cross increases monotonically from 820 to 1480 $\mathrm {\mu }{\rm m}$ from the weak to the strong breaker, respectively. The droplet diameter and velocity characteristics and the number of the droplets generated by the four mechanisms are found to vary significantly and the processes that create these differences are discussed.

Short-wave group forcing is a major driving mechanism of infragravity waves. The subharmonic response to wave group forcing approaches resonance in shallow water where the group velocity is equal to the shallow-water wave-propagating speed. Currently, there is a lack of understanding of the connection between the free and bound components of group-induced infragravity waves and the consistency among existing solutions for off- and near-resonance conditions in intermediate and shallow water. Here, a unified solution of group-induced subharmonics is derived based on Green's function for the first time. The new solution is valid for any resonance intensity and is able to describe group-induced subharmonic behaviour at all water depths consistently from a new angle. The proposed solution reduces to existing solutions for intermediate depth (Longuet-Higgins & Stewart, J. Fluid Mech., vol. 13, 1962, pp. 481–504; Zou, Phys. Oceanogr., vol. 41, 2011, pp. 1842–1859), shallow water and/or over a plane sloping beach (Van Leeuwen, PhD thesis, TU Delft, 1992; Schäffer, J. Fluid Mech., vol. 247, 1993, pp. 551–588; Janssen et al., J. Geophys. Res., vol. 108, 2003, p. 3252; Contardo et al., J. Phys. Oceanogr., vol. 51, 2021, pp. 1465–1487; Liao et al., J. Phys. Oceanogr., vol. 51, 2021, pp. 2749–2765). Unlike previous solutions, the Green's function-based solution describes all subharmonics as free subharmonics continuously radiated away from each point source in the group-induced forcing field determined by wave radiation stress gradients. The superposition of all these free subharmonics yields so-called bound subharmonics by previous studies due to group-modulated emission of each free subharmonic through the source field bound to the wave group. Therefore, this solution provides theoretical evidence that the group-induced subharmonic at any observation point is dependent on the surrounding radiation stress field and topography. Under full-resonance conditions in shallow water, downwave-propagating subharmonics excited at all source locations interfere with each other constructively; therefore, their superposed amplitude is proportional to the travel distance of wave groups. Combined with the conventional moving-breakpoint forcing model, the predicted amplitude of the subharmonic in the surf zone by the present solution is in good agreement with laboratory observations.

Discrete vortex rings impinging on concave hemispherical cavities were explored experimentally. Planar laser-induced fluorescence, two-dimensional particle image velocimetry and flow visualization techniques were employed. Five different ratios of vortex ring to hemisphere cavity radius ($\gamma$) were investigated, namely, $\gamma = 1/4,1/3,2/5,$$1/2, 2/3$. For $\gamma = 1/4,1/3, 2/5$, the geometric confinement of the primary ring due to the hemispherical cavity induced loop-like instabilities in the secondary ring, which led to head-on collision and ejection of the looped ends as they orbited the primary ring. As the hemispherical cavity decreased in diameter (increasing $\gamma$), the dynamics were altered significantly due to the increased generation of vorticity along the edge of the hemisphere. For $\gamma = 1/2$, vorticity produced at the edge/lip of the hemisphere ultimately disrupted the classical formation of a secondary vortex ring from the wall-bounded vorticity. For $\gamma = 2/3$, the primary ring and hemisphere radius were close enough in size that the interaction was dominated by direct impact of the primary ring with the lip of the cavity. The primary vortex ring produced a vortex ring at the lip of the hemisphere that ultimately separated from the cavity, orbited around the primary ring, and then self-advected in the direction opposite to the primary vortex ring trajectory. A detailed investigation of the dynamics provided.

Scallop patterns forming on erodible surfaces were studied historically using a linear analysis of the inner region of a turbulent boundary layer growing on a corrugated wall. Experimental observations show a phase shift between the shear stress at the wall and the wall oscillation that depends on the wavenumber. An ad hoc correction applied to the turbulent closure and due to Hanratty et al. (Thorsness et al., Chem. Engng Sci., vol. 33, issue 5, 1978, pp. 579–592; Abrams & Hanratty, J. Fluid Mech., vol. 151, issue 1, 1985, p. 443; Frederick & Hanratty, Exp. Fluids, vol. 6, issue 7, 1988, pp. 477–486) was systematically used to recover the reference experimental results. In this study, Reynolds-averaged Navier–Stokes (RANS) and direct numerical simulations (DNS) were performed and revealed the role of the Boussinesq assumption in the results obtained. We show that the Hanratty correction acts as a palliative to the misrepresentation of Reynolds stresses due to the use of the Boussinesq hypothesis. The RANS calculations based on a turbulence model using a second-order moment closure recovered the expected results obtained in the reference DNS calculations, in particular with respect to wall heat transfer. The analysis of these results highlights the critical importance of the anisotropy of the diagonal Reynolds stresses on the prediction of wall transfer under these conditions and their implication in the occurrence of scalloping.

This study investigates the coherence of turbulent fluctuations in a turbulent vertical natural convection boundary layer immersed in a stably stratified medium (turbulent buoyancy layer). A turbulent buoyancy layer of a fluid having a Prandtl number of $0.71$ at a Reynolds number of $800$ is numerically simulated using direct numerical simulation. The two-point correlations reveal that the streamwise velocity fluctuations are coherent over large streamwise distances, with the length scale of the streamwise coherence being greater than the boundary layer thickness. This is due to large-scale motions (LSMs), similar to the LSMs observed in canonical wall-bounded turbulence despite the stark differences in flow dynamics. Both high-speed (positive) and low-speed (negative) streamwise velocity fluctuations form LSMs, with their streamwise length scales increasing with increasing wall-normal distance. High-speed LSMs are composed of upwash flow with high temperatures, while low-speed LSMs are composed of downwash flow with low temperatures. Both high-speed and low-speed LSMs meander appreciably in the streamwise direction, with the degree of meandering being correlated with the sign of the spanwise velocity fluctuations. The LSMs exhibit coherence across significant wall-normal distances and contribute significantly to the turbulence production in the outer layer. Examining the one-dimensional energy spectra of the turbulent buoyancy layer shows that the LSMs are the dominant energy-containing motions, implying that the length scale of the energy-containing range is of the order of boundary layer thickness. Notably, wall-normal velocity, spanwise velocity and buoyancy fluctuations do not form LSMs with streamwise length scales comparable to streamwise velocity fluctuations.

We design an open-loop active flow control for separated flows around a plunging circular cylinder based on resolvent analysis. The cylinder is plunging at a Strouhal number of 0.36 and a Reynolds number of 500. A linear time-periodic system for control is obtained by linearizing the non-inertial incompressible vorticity equation in the cylinder-fixed frame about a time-averaged base flow. Using the Lyapunouv–Floquet transformation, the linear time-periodic system is transformed into a similar linear time-invariant system, whose resolvent is analysed to obtain an optimal actuating Strouhal number of 0.1464 for the transformed linear system. Simulations show that the active control with tangential actuations is capable of reducing the lift fluctuation by up to 25.7 % when the flow is actuated near the predicted harmonic and subharmonic frequencies.