The three-dimensional dynamic behaviour of a tip vortex generated by a NACA $66_2$-415 hydrofoil is investigated under wetted flow and cavitating conditions using time-resolved tomographic particle image velocimetry. Two main cavitation modes are studied, namely the breathing and double-helical modes. The time-averaged flow field consists of a system of two streamwise vortices for all three conditions. The shape of the tip vortex resembles that of the cavitation mode, instead of a circular cylinder. Multi-scale vortical structures are captured in the instantaneous flow field. The surface oscillation of the cavity contributes to the growth of Kelvin–Helmholtz instability over the tip vortex, leading to the onset of hairpin vortices. Stronger spanwise interaction between the tip vortex and flow separation of the hydrofoil is produced by cavitation, further intensifying the perturbation growth. The proper orthogonal decomposition analysis gives insight into the relationship between cavity oscillation and vortex instability. Two major types of unstable modes of the tip vortex are obtained, leading to serpentine centreline displacement and elliptical deformation motion. The selection of the dominant unstable mode is associated with cavity surface oscillation. For the wetted flow condition, the displacement mode dominates the growth of vortex perturbation, while for breathing mode cavitation, the most energetic unstable mode changes into an elliptical deformation pattern, the disturbance energy of which is negligible in the wetted flow condition. Consistency is found between the peak frequency of the deformation mode and the cavity resonance frequency, indicating the contribution of cavity oscillation to the disturbance growth and breakdown of the tip vortex.

Cavitation inception in the turbulent shear layer developing behind a backward-facing step occurs at multiple points along quasi-streamwise vortices (QSVs), at a rate that increases with the Reynolds number (Re). This study investigates the evolution of the unsteady pressure field and the distribution of nuclei within and around the QSVs. The time-resolved volumetric velocity in the non-cavitating flow is measured using tomographic particle tracking, and the pressure is determined by spatial integration of the material acceleration. Analysis in Eulerian and Lagrangian reference frames reveals that the pressure is lower, and its minima last longer within the QSVs compared with the surrounding flow. The intermittent low pressure regions, whose sizes and shapes are consistent with those of the cavities, are likely to be preceded by axial vortex stretching and followed by contraction. Such phenomena have been observed before in simulations of stretched vortex elements. For the same axial straining, the pressure minima last longer with increasing Re, a trend elucidated in terms of viscous diffusion of the stretched vortex core. The impact of nuclei availability is studied under ‘natural’ and controlled seeding. Owing to differences in the saturation level of non-condensable gas, the microbubble concentration in the shear layer decreases with increasing Re, in contrast to the rate of cavitation events. Minor differences in entrainment rate into the shear layer also do not explain the substantial Re effects on cavitation inception. Hence, the Re scaling of inception appears to be dominated by trends of the pressure field.

Ventilated cavitating flows are investigated via direct numerical simulations, using a coupled level set and volume of fluid method to capture the interface between the air and water phases. A ventilated disk cavitator is used to create the cavity and is modelled by a sharp-interface immersed boundary method. The simulation data provide a comprehensive description of the two-phase flow and the air leakage and vortex shedding processes in the cavitating flow. The mean velocity of the air phase suggests the existence of three characteristic flow structures, namely the shear layer (SL), recirculating area (RA) and jet layer (JL). The turbulent kinetic energy (TKE) is concentrated in the JL in the closure region, and streamwise turbulent fluctuations dominate transverse fluctuations in both SL and JL. Budget analyses of the TKE show that the production term causes the TKE to increase in the SL due to the high velocity gradients, and decrease in the JL due to streamwise stretching effects. Air leakage and vortex shedding occur periodically in the closure region, and the one-to-one correspondence between these two processes is confirmed by the velocity and volume fluid spectra results, and the autocorrelation function of the air volume fraction. Moreover, the coherent flow structures are analysed using the spectral proper orthogonal decomposition method. We identify several fine coherent structures, including $SL_{KH}$ induced by the Kelvin–Helmholtz instability, $SL_{out}$ associated with large-scale vortex shedding, $SL_{in}$ associated with small-scale vortex shedding, and $SL_{r}$ associated with upstream turbulent convection. The present study complements previous research by providing detailed descriptions of the turbulent motions associated with the violent mixing of air and water, and the complex interactions between different characteristic structures in cavitating flows.

The objective of this paper is to understand the effects of waves and vaporous cavitation upon the hydrodynamic and hydroelastic responses of a flexible surface-piercing hydrofoil, adding to the subcavitating results presented in Part 1. In general, the presence of a sufficiently large vaporous cavitation bubble facilitated the formation of a ventilated cavity, substantially reducing the angle of attack or speed required to induce fully ventilated flow, relative to subcavitating conditions. A new co-analysis procedure was used to isolate synchronous hydrodynamic modes and structural operating deflection shapes. Significant dynamic load amplification occurred when the resonant frequencies of the first twisting and second bending modes coalesced in both fully wetted and partially cavitating flows. The presence of waves did not diminish the effect of frequency coalescence, but did encourage intermittent lock-in with both leading edge cavity shedding and trailing edge vortex shedding at certain speeds. Partial cavity shedding typically had a negligible impact on the power spectral densities of structural motions because of incoherent cavity shedding. However, lock-in between the cavity shedding frequency and modal coalescence frequency led to shifting of the primary frequency peak, as well as amplified harmonics and interactions between the cavity shedding frequency and vortex shedding frequency. The transition from partially cavitating to fully ventilated flow caused sudden and large drops in the mean hydrodynamic loads and deformations, as well as substantial reductions in the intensity of the fluctuations.

This work describes the effects of waves on the hydroelastic response of a hydrofoil in fully wetted and ventilated flows. In the absence of vaporous cavitation (described in Part 2 of this paper series), shallow long-period non-breaking waves delayed ventilation inception because velocity fluctuations prevent the formation of a stably separated region of flow at the foil's leading edge. Aerated von Kármán vortex shedding occurred from the blunt trailing edge, producing vortex-induced vibration of the hydrofoil at a near-constant Strouhal number. Regular waves led to near sinusoidal oscillations of the load and deformations at the wave encounter frequency, while the mean response and the dynamic response at other frequency peaks corresponding to hydrodynamic and structural modes remained mostly unaffected. Significant dynamic load amplification was observed at a submerged aspect ratio of 2 for cases with low angles of attack because of coalescence between the second and third wetted structural modes; at high angles of attack, the amplitude of the load fluctuations and flow-induced vibrations reduced because energy was diverted away from the coalescence frequencies to the nearby vortex shedding frequency. In both calm water and wave conditions, transition from fully wetted to fully ventilated flow resulted in sudden and significant reduction in the load coefficients, as well as foil deflections. An impulse-like signature was observed in the time-frequency spectra during these transitions. In many of the cases, transition to fully ventilated flow also led to substantially reduced amplitudes in the load and deformation fluctuations.

It is known that the water entry of a body with a recessed, cupped nose can suppress the splash and air cavity typically observed for solid body entry (Mathai, Govardhan & Arakeri, Appl. Phys. Lett., vol. 106, 2015, 064101). However, the interplay between the captive gas in the cup, the cavity and the splash is quite subtle and has not been thoroughly explored. Here we study the cavity and splash dynamics associated with the vertical water entry of cups and find a variety of regimes over a range of Weber numbers ($We_D$) and dimensionless cup depths. Our parameter space spans a transition between slow-developing cavities with long closure times (low $We_D$) to fast-sealing cavities (high $We_D$). An important dynamic event is the evacuation of trapped gas from within the cup, which drives the ensuing cavity and splash behaviour. Through modelling, we predict the conditions for which the evacuating gas inflates a cavity that opens to the atmosphere versus inflating a submerged cavity that suppresses air entrainment from above the surface. We also compare our cup water entry findings to the impact phenomena observed for flat disks, which entrap gas on the front surface similar to cups. In doing so, we reveal the sensitivity of disk splash and cavity behaviour to impact angle, and show that disks share a common regime with cups, in which a thin splash quickly seals on the body. We deduce the mechanisms by which increasing cup depth delays the cavity seal time in this regime. These findings reveal that cups may in fact promote or suppress cavity growth, depending on the cup depth and impact conditions.

The collapse of a gas or vapour bubble near a non-porous boundary is directed at the boundary due to the asymmetry induced by the nearby boundary. High surface pressure and shear stress from this collapse can damage, or clean, the surface. A porous boundary, such as a filter, would act similarly to a non-porous boundary but with reduced asymmetry and thus reduced effect. Prior research has measured the cleaning effect of bubbles on filters using ultrasonic cleaning, but it is not known how the bubble dynamics are fundamentally affected by the porosity of the surface. We address this question experimentally by investigating how the standoff distance, porosity, pore size and pore shape affect two collapse properties: bubble displacement and bubble rebound size. We show that these properties depend primarily on the standoff distance and porosity of the boundary and extend a previously developed numerical model that approximates this behaviour. Using the numerical model in combination with experimental data, we show that bubble displacement and bubble rebound size each collapse onto respective single curves.

We propose a compressible multi-scale model that (i) captures the dynamics of both large vapour cavities (resolved vapour) and micro-bubbles (unresolved vapour), and (ii) accounts for medium compressibility. The vapour mass, momentum and energy in the compressible homogeneous mixture equations are explicitly decomposed into constituent resolved and unresolved components that are independently treated. The homogeneous mixture of liquid and resolved vapour is tracked as a continuum in an Eulerian sense. The unresolved vapour terms are expressed in terms of subgrid bubble velocities and radii that are tracked in a Lagrangian sense using a novel ‘$kR$-$RP$ equation’ (k, constant multiple; R, bubble size; RP, Rayleigh-Plesset). The $kR$-$RP$ equation is formally derived in terms of the pressure at a finite distance ($kR$) from the bubble while accounting for the effects of neighbouring bubbles; $p(kR)$ may therefore be either a near-field or far-field pressure. The equation exactly recovers the classical Rayleigh–Plesset and Keller–Miksis equations in the limits that $k$ and $c$ (speed of sound) become very large. Also, the results are independent of $k$ for a single bubble for all $k$, and for multiple bubbles when $kR < d$ (where $d$ denotes separation distance). Numerical results show this robustness of the model to the choice of $k$, which can be different for each bubble. The multi-scale model is validated for the collapse of a single resolved/unresolved bubble. Its ability to capture inter-bubble interactions is demonstrated for multiple bubbles exposed to an acoustic pulse. The model is then applied to a problem where resolved and unresolved bubbles co-exist. Finally, it is validated using a cluster of $1200$ bubbles exposed to a strong acoustic pulse. The results show the impact of the bubble cluster on the transmitted and reflected waves and the shielding effect where bubbles at the edge of the cluster shield the interior bubbles by dampening the incident acoustic wave.

Partial cavity flows forming on a NACA0015 hydrofoil are visualized using high-speed cinematography and time-resolved X-ray densitometry. These observations reveal the underlying flow features that lead to the cloud cavity shedding. Previous studies have reported that both near-surface liquid re-entrant flow and bubbly shock waves can serve as the mechanisms causing cavity pinch-off and cloud shedding. We identify both mechanisms in the current study. The cavity shedding frequency was also examined and related to the underlying flow dynamics. The probability of re-entrant flow or bubbly shock-induced shedding processes are quantified, and the likelihood of each mechanism is shown to be a function of both the cavitation number and the Mach number of the bubbly mixture within the separated region of the cavity. When the Mach number of the two-phase mixture in the cavity exceeds unity, shock waves become the dominant mechanism that lead to large-scale cavity shedding and cloud cavitation.

The physics and mechanism of sheet/cloud cavitation in a convergent–divergent channel are investigated using synchronized dynamic surface pressure measurement and high-speed imaging in a water tunnel to probe the cavity shedding mechanism. Experiments are conducted at a fixed Reynolds number of Re = 7.8 × 105 for different values of the cavitation number σ between 1.20 and 0.65, ranging from intermittent inception cavitation, sheet cavitation to quasi-periodic cloud cavitation. Two distinct cloud cavitation regimes, i.e. the re-entrant jet and shockwave shedding mechanism, are observed, accompanied by complex flow phenomenon and dynamics, and are examined in detail. An increase in pressure fluctuation intensity at the numbers 3 and 4 transducer locations are captured during the transition from re-entrant jet to shockwave shedding mechanism. The spectral content analysis shows that, in cloud cavitation, several frequency peaks are identified with the dominant frequency caused by the large-scale cavity shedding process and the secondary frequency related to re-entrant jet/shockwave dynamics. Statistical analysis based on defined grey level profiles reveals that, in cloud cavitation, the double-peak behaviours of the probability density functions with negative skewness values are found to be owing to the interactions of the re-entrant jet/shockwave with cavities in the region of 0.25 ~ 0.65 mean cavity length (Lc). In addition, multi-scale proper orthogonal decomposition analysis with an emphasis on the flow structures in the region of 0.25 ~ 0.65 Lc reveals that, under the shockwave shedding mechanism, both the re-entrant jet and shockwave are captured and their interactions are responsible for the dynamics and statistics of cloud shedding process.

Underwater stability of an air layer trapped in a micro-structure, plastron, is critical in drag reduction applications. Here, we investigate the wetting state and plastron stability of underwater superhydrophobic surfaces (SHS) under an intense acoustic drive. Flat surfaces and SHS are subjected to short acoustic pulses of different intensities. At low amplitude, the comparison between the results of various surfaces shows that plastron behaves like a water–air interface, whose presence can be detected from the phase of the reflected acoustic waves. At moderate intensity, a wetting transition towards a completely wetting state is observed and shown to be triggered by a sufficiently large acoustic radiation pressure. This wetting transition is well captured by a simplified model by balancing radiation pressure with the critical capillary pressure for the interface sliding. Cavitation clouds appear under strong excitation; their sizes and positions greatly depend on the surface acoustic boundary condition. For SHS in a Cassie–Baxter state (with an air layer), cavitation clouds appear at specific locations (from the solid surface) corresponding to the pressure anti-node of the transient standing wave generated by the reflection. This study unprecedentedly demonstrates the capability of acoustic waves to monitor and characterize plastron stability with low and moderate amplitudes, respectively.

We study the interaction between a laser-induced cavitation bubble and a submillimetre-sized water droplet submerged in silicone oil. High-speed imaging reveals the pathways through which droplet fragmentation occurs and three distinct regimes of bubble–droplet interaction are identified: deformation, external emulsification and internal emulsification. We have observed that during the bubble collapse, the droplet elongates towards the bubble, which acts as a flow sink pulling on the droplet. For silicone oils with higher viscosity, the droplet jets into the cavitation bubble and forms a satellite water droplet in the continuous oil phase. In contrast, for lower-viscosity oils, the droplet encapsulates the collapsing bubble as it jets inside and undergoes multiple cycles of expansion and collapse. These internal bubble collapses create tiny oil droplets inside the parent water droplet. The kinematic viscosity of the silicone oil, maximum bubble diameter and centre-to-centre distance between the bubble and the droplet are varied. The regimes are separated in a parameter space set up by the non-dimensional distance and a cavitation Reynolds number.

The hydrodynamic cavitation in semidilute solution flows of a flexible polymer additive in water was experimentally explored in a mesoscale converging–diverging nozzle to elucidate the cavitation reduction effects of polymer additives. Rheological measurements demonstrated that polymer solutions were shear-thinning, with infinite viscosities larger than pure water. The polymer additives significantly mitigated the intensity of cloud cavitation and the growth of violent cavity structures in the tested solution concentrations. Under conditions of supercavitation, the tested polymer solutions could not suppress the growth of large structures but showed a reduction in the population of cavitation bubbles. The temporal evolution and spatial variation of cavitation structures in different concentrations were captured using high-speed imaging. Statistical analysis of the images showed that polymers reduce cavitation via three main mechanisms. (1) The longitudinal expansion of cavities downstream is attenuated relative to the pure water. The streamwise distribution of vapour-ratio fluctuations was flattened, and its peak was shifted upstream in the solutions. (2) Mean collapse and growth rate of cavitating bubble pockets and their fluctuations were noticeably relaxed by polymer additives. For a 400 p.p.m. solution (parts per million (p.p.m.)), a reduction of 65 % was measured relative to pure water flow at the highest tested flow rate. (3) Spectral analysis of the downstream pressure indicated that the shedding frequency at the cavitation inception was reduced as the solution's concentration increased. This reduction was as high as 70 % for a 400 p.p.m. solution. These results highlight the strong interplay between polymer additives and the generation of cavitation-related structures.

A core unresolved challenge in rectified diffusion is the lack of predictive models, which allow accurate modelling of the growth of bubbles over millions of cycles of oscillation. The spherically symmetric problem involves two asymptotic regions in space for the convection and diffusion of the dissolved gas in the liquid, an inner region adjacent to the bubble and an outer region. It also involves three time scales: the shortest time scale associated with the bubble oscillation, the intermediate and the longest time scales for the mass transport across the bubble interface and the outer region, respectively. The problem is analysed systematically using matched asymptotic expansions in space and multi-scales in time, the large parameter being the Péclet number for mass transfer. A tractable mathematical model is formulated for rectified diffusion. It comprises the Rayleigh–Plesset equation for the spherical oscillations of the bubble on the shortest time scale, an ordinary differential equation for the mass of gas in the bubble on the intermediate time scale and a diffusion equation on the longest time scale and the long length scale. Predictions of this simplified model have excellent agreement with experimental results for a single spherical bubble in the bulk over millions of cycles of oscillation. A parametric study is undertaken with several new phenomena/features observed for acoustic frequencies in the range 12 to 62 kHz, acoustic field $0.12$ to $0.22$ bar and the initial gas concentration in the liquid being $0.95$ to $1.0$ of the saturation concentration.

This work reveals that the dynamic response of a spherical cap bubble in contact with a rigid wall depends on the effective contact angle at the instant prior to collapse. This parameter allows us to discriminate between two regimes in which the mechanisms of interaction between the collapsing bubble and its surrounding medium differ significantly: when the contact angle is smaller than $90^{\circ }$, a classical jet directed towards the wall is observed, whereas if the initial contact angle is larger than $90^{\circ }$, an annular re-entrant jet parallel to the wall appears. We show that this change of behaviour can be explained using the impulse potential flow theory for small times, which shows the presence of a singularity on the initial acceleration of the contact line when the contact angle is larger than $90^{\circ }$. Direct numerical simulations show that although viscosity regularises the solution at $t > 0$, the solution remains singular at $t=0$. In these circumstances, numerical and experimental results show that the collapse of flat bubbles can eventually lead to the formation of a vortex ring that unexpectedly induces long-range effects. The role of the bubble geometry at the instant of maximum expansion on the overall collapse process is shown to be well captured by the impulse potential flow theory, which can be generalised easily to other bubble shapes. These results may find direct application in the interpretation of geophysical flows as well as the control and design of biomedical, naval, manufacturing and sonochemistry applications.

The jetting behaviour and migratory characteristics of a laser-induced cavitation bubble in a rectangular channel are investigated both experimentally and numerically, for various combinations of the geometric and physical parameters of the system. High-speed photography is used to visualize the temporal development of the bubble shape, the formation of liquid jets during bubble collapse, and the bubble displacement in contact with the sidewalls of the channel during two oscillation cycles of the bubble. The bubble profiles, pressure contours and velocity vectors ambient to the bubble are obtained through numerical simulation results by using an Eulerian finite element method with a compressible liquid impact model. The jetting behaviour of the bubble varies between single jet formation and the formation of three liquid jets directed towards each wall of the channel. The numerical calculations indicate that the liquid jets directed towards the sidewalls of the channel reach maximum velocities of 100 m s−1 while the peak velocity of the liquid jet directed towards the channel endwall is about 55 m s−1. A small bubble generated close to a sidewall of the channel develops only a single inclined jet during collapse. Such jets can reach velocities of up to 110 m s−1. A bubble displacement in contact with the sidewalls of the channels of 350 μm was observed during the first two oscillation cycles for a bubble with a maximum diameter slightly smaller than the height of the channel. The results of our investigations are compared to previous results obtained in similar configurations.

We study the dynamics of a cavitation bubble near beds of sand of different grain sizes. We use high-speed imaging to observe the motion of the bubble and the sand for different values of the stand-off parameter $\gamma$ (dimensionless bubble-boundary distance) between 0.3 and 5.3. Compared with a rigid boundary, we find that a granular boundary leads to bubbles with shorter lifetimes and reduced centroid displacements. Above $\gamma \approx 1.3$, the behaviour of the bubble is almost independent of the granularity of the sand. When the stand-off parameter lies between $0.6$ and $1.3$, a mound of sand develops beneath the bubble, which can force the latter to assume a conical shape as it collapses. For $\gamma \lesssim 0.6$, the bubble develops a bell-shaped form, leading to the formation of thin and surprisingly fast micro-jets ($v_{jet} > 1000\,{\rm m}\,{\rm s}^{-1}$). Moreover, between $\gamma \approx 1.3$ and $\gamma \approx 0.3$, granular jets erupt from the sand surface following the bubble collapse. We additionally develop a simple numerical model, based on the boundary integral method, to predict the dynamics of a cavitation bubble near a bed of sand, which we replace by an equivalent liquid. The simulations are remarkably consistent with experimental observations for values of $\gamma$ down to $1.3$. We also show how the anisotropy parameter $\zeta$, a dimensionless version of the Kelvin impulse, can be adapted for the case of a nearby bed of sand and predict the displacement of the bubble centroid for $\zeta \lesssim 0.08$.

A mathematical model for the simulation of spherical bubble dynamics in binary alkane mixtures is presented. Detailed heat and mass transfer and phase transition are resolved, and air absorption and desorption are considered. As example mixtures, high-volatile heptane and low-volatile dodecane are investigated. The low-volatile component shows a convective counter-intuitive mass transport opposite to the diffusion flux. A staggered mass flux of heptane and dodecane is associated with a local segregation of mixture components in the surrounding liquid and a distinctive species distribution within the bubble. A comparative study with a pseudo-fluid where alkane species interdiffusion is absent and whose components cannot segregate reveals that for an oscillating bubble, the local segregation hardly affects bubble dynamics, while for a continuously growing bubble in a superheated liquid, the growth rate is considerably affected, particularly for a low heptane mixture percentage. This study demonstrates limitations of treating mixtures by a single-component surrogate fluid and may serve as a starting point for the development of multi-component cavitation models for computational fluid dynamics applications in real fluid mixtures as, e.g. fuels or hydraulic oils.

The dynamics of cloud cavitation about a three-dimensional hydrofoil are investigated experimentally in a cavitation tunnel with depleted, sparse and abundant free-stream nuclei populations. A rectangular planform, NACA 0015 hydrofoil was tested at a Reynolds number of $1.4\times 10^{6}$, an incidence of $6^{\circ }$ and a range of cavitation numbers from single-phase flow to supercavitation. High-speed photographs of cavitation shedding phenomena were acquired simultaneously with unsteady force measurement to enable identification of cavity shedding modes corresponding to force spectral peaks. The shedding modes were analysed through spectral decomposition of the high-speed movies, revealing different shedding instabilities according to the nuclei content. With no active nuclei, the fundamental shedding mode occurs at a Strouhal number of 0.28 and is defined by large-scale re-entrant jet formation during the growth phase, but shockwave propagation for the collapse phase of the cycle. Harmonic and subharmonic modes also occur due to local tip shedding. For the abundant case, the fundamental shedding is again large-scale but with a much slower growth phase resulting in a frequency of $St=0.15$. A harmonic mode forms in this case due to the propagation of two shockwaves; an initial slow propagating wave followed by a second faster wave. The passage of the first wave causes partial condensation leading to lower void fraction and consequent increase in the speed of the second wave along with larger-scale condensation. For a sparsely seeded flow, coherent fluctuations are reduced due to intermittent, disperse nuclei activation and cavity breakup resulting in an optimal condition with maximum reduction in unsteady lift.