## Abstract

When a steep breaking wave hits a vertical sea wall, in shallow water, a rapidly ascending planar jet forms. This jet is ejected with high acceleration due to pressure created by the violent wave impact on the wall, creating a so-called ‘flip-through’ event. Previous studies have focused on the impulsive pressures on, and within, the wall and on the velocity of the jet. Here, in contrast, we consider the formation and break-up of the jet itself. Experiments show that during flip-through a fluid sheet, bounded by a rim, forms. This sheet has unstable transitional behaviours and organizing jets; undulations in the thickness of the fluid sheet are rapidly amplified and ruptured into an array of vertical ligaments. Lateral undulations of the rim lead to the formation of finger-jets, which subsequently break up to form droplets and spray. We present a linear stability analysis of the rim–sheet systems that highlights the contributions of rim retraction and sheet stretching to the break-up process. The mechanisms for the sequential surface deformations in the rim–sheet system are also described. Multiple, distinct, instability modes are identified during the rim deceleration, sheet stretch attenuation and rim retraction processes. The wavenumbers (and deformation length scales) associated with these instability modes are shown to lead to the characteristic double peak spectrum of surface displacement observed in the experiments. These mechanisms help to explain the columnar structures often seen in photographs of violent wave impacts on harbour walls.

## 1. Introduction

When a shoaling wave crest approaches a vertical wall, the wall prevents the forward flow of water causing the water level at the wall to rise rapidly. The surface between the rising wave trough and advancing crest converges rapidly, and often focus at a point on the wall (figure 1), resulting in violent wave impact—the so-called flip-through event [1,2]. The thin film-like vertical jet formed at the flip-through is projected upward at high acceleration [3].

In engineering, most attention on this wave–wall interaction has concentrated on the impact pressures acting on the wall during either sloshing impacts [3] or breaking wave impacts [4,5]. The pressure impulse during the flip-through impact has been derived theoretically by Cooker & Peregrine [1,2], providing a mathematical interpretation of underlying dynamics before the start of uprush.

The normal theoretical description assumes both potential flow and homogeneity along the wall, allowing the problem to be simplified by removing the third, transverse, coordinate direction [4]. We have found, however, that the high pressure gradient along the wall, resulting from the impact, accelerates the fluid vertically ejecting a planar thin jet, which follows with inherent transverse deformations (see figure 2 and electronic supplementary material, movie S1); a horizontal straight rim initially forms along the free edge of the planar jet and subsequently evolves cusp-like formations with simultaneous undulations in the thickness of the film bounded by the rim. The undulations in the rim are rapidly amplified to produce typical finger-shaped jets on the rim as the sheet ruptures. The finger jets are then fragmented into large numbers of droplets forming a spray. The formations of sea spray observed during wave impacts on sea walls may be interpreted using the unstable behaviours of these uprising jets. Such an interpretation is presented here for the first time.

The free-surface dynamics through fingering and breaking up processes have been observed in breaking ocean waves, crown splash and liquid sheet fragmentation.

### (a) Breaking waves on beaches

On shallow water coasts (without walls), depth-induced breaking waves have been also found to produce secondary finger jets which are fragmented into droplets at wave splashes. When a breaking wave crest overturns and impacts onto a forward water surface, a secondary jet is rapidly formed and is projected forwards. The initial, two-dimensional, spanwise vorticity produced along the inner surface of the overturning jet becomes unstable and the change in orientation results in multiple pairs of counter-rotating vortices that are stretched obliquely downward beneath the surfaces of the secondary jet [6]. The surface just above these vortex pairs is entrained into the interior of the water to create a convergent free surface flow, resulting in the formation of a transverse array of longitudinal scars on the wave surfaces [7]. In the secondary jet, stretching causes the counter-rotating vortices to be intensified within the jet, thus the vortex-induced scars on the jet surfaces are deepened and finally penetrate the jet forming multiple finger jets [8]. The fingers are stretched along the jet axes, breaking up into droplets via capillary instability. A series of these studies concludes that the initial shear instability manifested in breaking waves triggers the transverse deformation of the jet, evolving into the fingers via vorticity dynamics.

### (b) Crown splashes

Crown splashes are also known to produce fingers on an undulating circular rim as a result capillary instability during the retracting behaviour of the rim (see the review by Eggers & Villermaux [9]). In this event, a droplet impact initially induces a cavity on receiving liquid and a circular fluid sheet rising with rapid expansion. A circular rim formed along the upper edge of the sheet is pulled back by surface tension during the retracting process. The volume of the rim increases until unstable undulations in the rim grow sufficiently causing the crown structure with the finger jets aligned along the rim. Many investigations of the unstable motion of the rim and the initiation of finger jets have been performed [10–12] and of the unsteady behaviour of the stretching rim [13,14]. The recent study by Agbaglah *et al.* [12] concludes that retracting behaviour initially governs the rim instability for short duration in the Rayleigh–Plateau (RP) mode, then Rayleigh–Taylor (RT) instability starts to contribute as the fingers are initiated.

### (c) Liquid sheets

Capillary sheet dynamics have been widely studied since theoretical interpretations of the free edge and radically expanding sheet by Taylor [15,16]. Mechanisms for the atomization of the sheet through a combination of interactive capillary, rim and centrifugal dynamics have been investigated for more complex rim–sheet system [17,18]. Bremond *et al.* [18] found that the capillary waves travelling on the sheet to the free edge induce amplification of the transverse thickness modulations, leading to the formation of ligaments perpendicular to the sheet edge. Yarin & Weiss [19] showed the formation of a cusp-shaped rim is due to capillary discontinuity waves propagating on the sheet.

Strong stretching of the planar jet is known to modify the capillary response. Longuet-Higgins [20] presented another possible mechanism for the longitudinal disintegration of pre-splash overturning jets via modification of the capillary orientation in stretched jet surfaces; capillary waves with transverse crests emerged on a curved free-falling overturning jet. Change in orientation by gravity-induced stretch amplifies the capillary amplitude, causing the lateral disintegration of the falling jet.

These previous findings for breaking waves on beaches, crown splashes and thin liquid sheets provide theories for the dynamics of the flip-through jets induced by wave impacts on vertical walls. In this study, the mechanical and geometrical features of the finger and ligament formations on the uprushing jet are experimentally characterized using these analogies. The mechanical stability of the flip-through jets is also considered using a linear stability analysis for the rim–sheet-coupled system using a long-wave approximation, and comparisons with the experimental results are made.

The paper is organized as follows. In §2, the experimental set-up and conditions are explained. Section 3 discusses the geometric features of the deformations of the rim and sheet observed experimentally following the wave impact. In §4, the unstable behaviour of the rim–sheet system is discussed, using linear stability analysis, as a model for the jet formed at flip through. Section 5 explains the observed mechanisms to deform the jets during the flip-through event in terms of the transitional instabilities amplified on the system. Finally, the results are summarized in §6.

## 2. Laboratory experiment

In this section, we explain the measurement system set-up for the water wave experiments comprising a laboratory wave flume with the vertical wall; an optical system with a high-speed camera; and the wave measurement and control system.

The experiments were performed in a 24 m long wave flume, with a width of 0.6 m and depth of 1.0 m, fitted with a piston type wave maker with active wave absorption. A transparent, acrylic, rectangular, box 0.2 m long and 0.6 m wide with a height of 1.0 m was installed on the *x*-axis, the nearside of the flume in the transverse, *y*-axis and at the still water level in the vertical *z*-axis. The still water depth at the origin *x*=0 was 100 mm.

The resulting wave impact pressure is very sensitive to the incident wave shape at the impact [4], and consequently the surface behaviour of the vertical jets also depends strongly on the surface shape of the wave face (or local wave steepness). In general, progressive waves with mild steepness reflect off the wall and behave like standing waves at the wall, while steeper waves produce more violent impacts causing highly accelerated thin up-rushing fluid jets. The post breaking wave impact provides more complex fluid flows involving entrained air bubbles and induced turbulence. Therefore, in the experiments, the position of the vertical wall, relative to the breaking point, was taken to be a parameter used to define the impact modes of shoaling waves; During the experiments, the model breakwater was located at *x*=−400,−300,−200,−100 mm (in the pre-breaking region), 0 mm (for the flip-through mode), and 200 mm (for the post-breaking region).

A rectangular blue light-emitting-diode (LED) panel (200×100 mm) was set behind the transparent front wall of the model breakwater and used to illuminate the dyed water passing over it. An 8-bit high-speed video camera recorded the illuminated area using a 45° angled reflector (figure 3). The region illuminated by the LED panel was also traversed vertically from 50 mm above the still water level to +300 mm at 50 mm intervals to allow the vertical evolution of the jets to be examined. The water in the flume was coloured using Uranine dye (Sodium Fluorescein), to enhance contrast of the liquid region. Uranine excites with blue, 436 nm, light and fluoresces with a green, 530 nm, light. A low-pass optical filter (less than or equal to 450 nm) was fitted to the camera, to eliminate the fluorescent emissions from liquid and allow only the blue component, from the back light, to be recorded. The concentration of the dye, 0.11 mg l^{−1}, was chosen to ensure that there would be no light transmission across a 0.25 mm thick fluid film. This set-up ensures that the image brightness on the liquid area roughly indicates the thickness of the sheet as the transmitted light intensity through the film proportionally decreases with the film thickness. Images (8-bit, 1280×1024 pixel) were recorded at 500 Hz with an exposure time of 0.125 ms and stored on a PC connected to the camera as uncompressed bitmaps.

Throughout the experiments, monochromatic waves with the incident wave height of 146 mm and a wave period of 1.9 s were used. Two wave gages were installed in the constant depth section of the flume, at water depth *h* of 275 mm, to determine the incident wave height (figure 3). When the first wave passed the shoreward wave gage, a TTL signal was sent to the high-speed video camera to start recording, this synchronizes the impact phase of each of the 20 trials performed for each experimental case to allow a statistical analysis of the results.

In a post-processing procedure, noise on the acquired images was reduced using median filtering. Image coordinates were transformed to real coordinates using a linear image transformation, providing quantitative measures of the liquid surface on images at 0.10 mm per pixel resolution.

## 3. Experimental results

This section describes the features observed during the experiments in the three impact regimes (pre-breaking, flip-through and post breaking). The transverse deformations of the jets are characterized in terms of wavenumber spectrum.

### (a) Wave impacts

The dynamics of the up-rushing jet depend both on the relative location of the sea wall and the point at which the wave breaks. Clearly, the initial vertical acceleration of the jet resulting from the wave impact depends on the incident wave steepness immediately prior to impact (see detailed in review by Peregrine [4]). If the wall is far from the breaking zone, linear, sinusoidal waves are reflected by the sea wall forming the characteristic pattern of standing waves. Steeper, shoaling, waves, however, cause rapid changes in dynamic pressure at the wall, accelerating the surface upward during the wave–wall interaction (see phases (*a*)–(*c*) on the top panel of figure 4). The vertically accelerated water surface at the wall forms a planar thin film-like jet extending from the standing wave crest as shown in phase (*d*). In the flip-through case, a nearly vertical wave face progressing towards the wall focuses at a point on the wall with the forward trough lifted up (see phase (*c*) of figure 1). A high-pressure gradient occurring at the focus point ejects the water sheet extending from the standing wave crest (phase (*d*) of figure 1, and (*a*–*d*) of figure 5).

In the case of broken wave impact, the overturning jet entraps air prior to impact (see phases (*a*,*b*) on the bottom panel of figure 4). During the impact, the trapped air is squeezed before finally breaking up into large numbers of bubbles. The resulting aerated water spray is dispersed over a wide area. The entrapped and entrained bubbles significantly change the dynamics of the wave impact; the compressibility of air causes pressure oscillations on the wall and alters the duration of the pressure rise, maximum pressure and the sound speed [4,5]. The combination of these multiple effects of air mixture yields very complex fluid motion in the jet.

Peregrine [21] provided a mathematical model of a splash from a thin layer of water. He interprets the projection of a jet at an angle with respect to the fluid layer by considering the conservation of mass and momentum using Bernoulli’s theorem and including both the impact and projections points. The expansion of the crown formed by droplet impacts can be described in terms of the propagation of a discontinuity wave at the projection point on the layer streams [19]. Roisman & Tropea [22] extended this model to derive a solution of the crown shape.

Cooker & Peregrine [2], on the other hand, derived a theoretical solution for the fluid flow in a rectangular wave impact and found that the vertical velocity (*v*_{1}) on the free-surface become infinite at the wall location (*x*=0); *U*_{0} is the impact velocity normal to the wall and *H* is the distance from the bottom to the wave crest (figure 5*e*). Assuming that the upward stream in a thin fluid layer, with thickness *h*_{1}, on the vertical wall above the free-surface of the rectangular wave occurs at the inception of the impact, despite the singular velocity at the wall, the finite mean velocity may be found by explicitly integrating *v*_{1} over the layer thickness; *h*_{1} to *h*_{2} and velocity from *e*), the mean velocity vector of the jet, *h*_{B}=*h*_{1}+*h*_{2}, can be found. In this case of static upper film with *h*_{B}≈0.1 mm (see §4f) and the thinnest lower film of one order thinner than *h*_{B} above the free-surface, *h*_{1}≈0.01 mm, the approximate measures of the mean velocity vector and splash angle are estimated to be *α*≈23°, respectively.

Figure 5*a*–*d* shows the local orientation of the uprising jet formed during the flip-through impact. The jet is ejected upward at angles with respect to the vertical wall that are consistent with the inclination angle approximated by the splash model. This indicates that the uprising fluid moves upward without contacting the wall during the initial jet projection. The flip-through jet may therefore be treat as a free jet.

Figure 6 shows the mean upward velocity of the jet (measured using image analysis) plotted against the location of the wall at three different heights above the still water level. The vertical velocity decreases with increasing height as drag force and gravity decelerates the water sheet. The maximum vertical velocity (approx. 6.7 ms^{−1}) occurring at flip-though (*x*=0) is more than ten times the maximum vertical velocity estimated by the small-amplitude linear wave theory [23] for the standing wave (approx. 0.6 ms^{−1}). Although an exact comparison of the observed velocity with the analytical predictions of Cooker & Peregrine [2] is impossible because of the singularity at the wall, the predicted mean velocity of the jet is of the same order of the mean velocity of the observed jet. We note that the velocity is also comparable with that reported by Bruce *et al.* [24] for random wave impacts on a 10 : 1 battered wall. As the breaking point moves seaward, the jet velocity falls off rapidly.

### (b) Evolution of jets at flip-through

In the experiments no corrugations or irregularities are observed in the wave crest which could lead to transverse changes in velocity during the impact. However, an uneven transverse surface can be observed in the up-rushing sheet together with a defined rim following the impact.

The evolution of the sheet as it initially forms finger jets before rupturing and breaking up into spray is shown in figure 7 (see also electronic supplementary material, movie S1). In the early stages of figure 7*a*,*b*, a rim is observed to form at the leading edge of the sheet-like jet. Figure 7*b* clearly shows a striped pattern below the rim–indicating transverse undulations of sheet thickness, these may be due to the capillary re-orientation causing longitudinal disintegration of the thinning jet [20]. As the jet rises (figure 7*b*–*d*), the transverse undulations near the rim are amplified forming cusp-like structures before evolving into a regular pattern of finger jets. During this stage of evolution, the sheet is being stretched and consequently thins.

The amplified undulations of sheet thickness rupture in the fully stretched sheet, resulting in holes that are then elongated vertically (figure 7*c*,*d*), finally an array of vertical ligaments is formed which merged with the deformed rim (figure 7*d*,*e*). The rim–ligament system detaches from the standing wave crest in figure 7*e*,*f*, before finally disintegrated into individual droplets through capillary instability. The fact that a wetted wall is not observed after preceding jets have passed in electronic supplementary material, movie S1, further indicates that there is no contact of the jet with the wall.

Saruwatari *et al.* [8] found that the mechanism for forming fingers in jets splashing onto a still water is governed by vorticity dynamics within the sheet not by surface tension. They failed to find, however, any evidence of rim formation at either the leading edge of the jet nor the ruptured sheet (as in figure 7), suggesting the mechanisms observed above are different from those for splashing jets. Consequently, vortex dynamics may be irrelevant in this issue.

The transverse rim instability emerged in the early stages (figure 7*b*–*d*) is visually analogous to that observed in the crown splash, which may also be interpreted by considering the retracting dynamics and the capillary effect. The regular deformations and break-up of the sheet, formation of the longitudinal ligaments and the detachment of the rim from the sheet have, however, never been reported in crown splashes. The simultaneous amplification of the undulations in sheet thickness, which may be triggered by destabilization of the stretched, thinning sheet during re-orientation of capillary waves [20], suggests the dynamics of the coupled rim–sheet system governs the break-up mechanism for flip-through jets.

### (c) Formation of jets during pre- and post-breaking impacts

Figure 8 shows a series of images of the resulting sheet at *z*=100 and 200 mm above the still water level in pre- and post-breaking cases. When the wall is seaward of the breaking point, a standing wave with a uniform transverse wave crest forms. As the wave runs up on the vertical wall, a sheet of water is ejected upwards, which in this case is in contact with the wall. In figure 8*a*,*b* with the wall 200 mm seaward of the breaking point, as the sheet emerges small disturbances appear along the edge of the sheet, which are amplified as the sheet stretches to form regular patterns of fingers. In the non-breaking case, where the jet contacts the wall, the shear between the fluid and the wall that also affects the deformation of the jet, which should be treat as a moving contact line problem. Troian *et al.* [25] identified unstable transverse behaviour of a spreading film on a slope to interpret the finger formation of the film driven by capillary, gravity and shear balance, which may be relevant to the local deformation of the pre-breaking jet in the deeper water region.

When the wall is closer to the breaking point at *x*=−100 mm, a quasi-flip-through process is observed (figure 8*c*,*d*). In this case, a rim with lateral undulations is clearly identifiable on the sheet at *z*=100 mm, while the neck of the jet (associated with the standing wave component) moves up the wall. As the sheet rises, the rim becomes detached from the sheet and ligaments are formed as the sheet ruptures. These features indicate that, in this case, the motion of the preceding sheet can be modelled as a free jet without wall contact.

In the post-breaking case, an air pocket is trapped by the overturning jet and is compressed and collapses into many bubbles when the broken wave hits the wall (see the bottom sequence in figure 4). The aerated fluid forms many very complex and irregular jets, which are dispersed in many directions, as a result of the combined effects of aeration and turbulence (figure 8*e*,*f*). Large numbers of the spray jets are produced during the initial stage of the impact with some jets evolving into ligaments, which successively break up into droplets due to the capillary instability. Although it is interesting to discuss the complex fluid motions in this region, many unknown mechanical factors, such as compressibility effects and modifications of pressure responses owing to aeration [4,5], surface-vorticity interactions [8] in addition to the capillary and sheet dynamics, need to be understood for identifying the broken wave impacts, which should be left for future investigations. In this paper, we therefore concentrate on the mechanism of surface deformation and break-up for the flip-through case.

### (d) Geometric features of the flip-through jets

In order to quantify the observed deformations of the uprising jets, the edges of liquid areas on the images were detected by an active contour model based on an energy fitting algorithm using a level-set iteration [26]. The detected boundaries for liquid jets and sprays were defined separately; spray droplets are defined as having a closed boundary, while the jet boundary must touch the edge of the image (figure 9*a*). Liquid which is out of the focal plane of the camera and which has the maximum absolute gradient of image intensity along the liquid boundaries lower than a given threshold, is removed from ensemble statistics.

An integral length *F*(*y*) is useful for quantifying the transverse undulations of the jets. *F*(*y*) is defined by counting the pixels in the *y*th column which are inside a jet (figure 9); pixels inside spray droplets are explicitly excluded from the count. Figure 9 shows *F*(*y*) plotted at three different times, following impact for the flip-though case at *z*=100 mm (figure 7*b*); the corresponding photographs (*a*–*c*) are shown above the graph. Although a double-valued surface profile with respect to *y* may smear *F*(*y*) somewhat, we find *F*(*y*) consistently defines the transverse locations of the cusped rims and fingers (indicated by broken lines in the sequential images (*a*–*c*)) and this is used for estimating the major intervals of the rim deformations.

The temporal behaviour of the transverse variations of the integrated lengths are analysed using the Fourier spectrum *S*(*K*) to determine the length scales of the deformations (figure 10). Here, the dimensional wavenumber *K*=2*π*/*L* is used, where *L* is the transverse wavelength (in mm). It should be noted that the lowest wavenumber component, corresponding to the mean surface location, is not shown so as to emphasize higher wavenumber components. A prominent spectrum band widens as the breaking point approaches, i.e. *K*<1.0 mm^{−1}, furthest from the breaking point (*x*=−200 mm) and is widest, 0.1<*K*<2.0 mm^{−1}, at flip-through (*x*=0 mm). At *x*=0 mm, two significant peaks are observed in the spectrum at *K*≈ 0.26 and 0.67 mm^{−1} (corresponding to about 24.2 and 9.4 mm of wavelengths). This suggests two distinctive fluctuation modes triggered by different instability mechanisms govern the formation of the transverse deformations. The observed spectrum is compared later with linear instability growth.

Figure 11 shows a sequence of the mean relative complement image brightness (CIB), averaged over the transverse, *y*, directions plotted against the *z* coordinate. As previously discussed (see §2), the CIB is proportional to the thickness of the fluid sheet, consequently the CIB=1 if the backlit fluid sheet is thicker than 0.25 mm and will be zero where there is no fluid in-front of the back light. Figure 11 shows a region of relatively high mean CIB appearing behind the leading edge, due to the presence of the deformed rims, followed by a thinner sheet region, with a correspondingly lower CIB. As the jet rises the sheet, between the wave crest and the rim, elongates and thins. This geometric feature, a thinning sheet with approximately constant thickness, is used as the basis of the rim stability analysis presented in §4.

### (e) Flip-through versus crown splash

The distinct behaviours of splashing jets, during wave impacts, depend on the relative location of the breaking point and the wall and may be characterized by analogy with droplet impact on a solid wall. Rioboo *et al.* [27] provide a description of the possible outcomes of droplet impact on a dry wall. The deposition process is characterized, for relatively low impact speed and small droplet size, by a radially spreading lamella which forms during the impacting process (figure 12*a*). At higher impact velocity, the corona rim and sheet are radically ejected at an inclination angle with respect to the bottom wall (figure 12*b*). As the corona (or crown) grows azimuthal stretching and expansion modify the capillary instabilities on the rim [28]. Droplet impacts on a thin liquid film typically produce identical crown splashes with a specific splash angle [29].

In the case of the splashing jet (figure 12*c*,*d*), the underlying process can also be split into two distinct types. In the pre-breaking case, a laterally uniform planar sheet moves up the vertical wall (figures 8*a*,*b* and 12*c*). This type is directly comparable to the deposition type of the droplet impact, but with the thinning sheet spreading upwards rather than radially. In a similar way, the flip-through jet can be considered as a planar version of the axisymmetric crown splash. Here, the wave impact ejects a fluid sheet, bounded by a rim at an acute angle to the wall (see figures 5 and also 12*d*). Unidirectional longitudinal stretching of the sheet results in rapid thinning with transverse undulations. As the sheet stretches, these lead to longitudinal ruptures forming a vertical array of ligaments and detachment of the rim from the sheet (figure 7). This mode is never observed in crown splashes with radical expansion. The observation that the unstable behaviour of the rim sheet system is dominated by longitudinal stretching is the basic assumption for the present mathematical model.

### (f) Scaling the jet motion

The rim size and the sheet thickness observed in the experiments can be roughly estimated from the back-light image intensity (see §2). The rim radius, *a*_{e}, is in the range of 1.0–1.5 mm during the flip-through event, while the sheet thickness, *h*_{e}, just before rupturing is less than 0.1 mm. Consequently, the representative lengths, *a*_{e}≈ 1.0 mm and *h*_{e}≈ 0.1 mm have been used for scaling the dynamics of the flip-through jets.

Based on these thicknesses, a characteristic velocity and time scales can be estimated. The retracting behaviour of the sheet is associated with the Taylor–Culick velocity, ^{−1}. Here, *ρ*_{l} is the density of liquid and *γ* is the surface tension. While this is less than the initial velocity of the jet (approx. 6.7 ms^{−1}, figure 6) it is of a comparable order of magnitude. The time scale associated with the capillary motion of the sheet is *τ*_{ch}≈1.2×10^{−4} s and *τ*_{cr}≈3.1×10^{−3} s. At the initial stage of jet formation, the rim and sheet thicknesses are comparable, so consequently *O*(*τ*_{cr})∼*O*(*τ*_{ch}). With simultaneous onsets of the rim growth and sheet thinning, however, the increasing difference between *τ*_{cr} and *τ*_{ch} may lead the sequence of rim and sheet behaviour observed with time lags shown in figure 7. Transverse undulations in the sheet thickness are first observed (figure 7*a*). As the sheet thins, there is rapid growth in these undulations eventually leading to rupture (figure 7*b*,*c*), while slower transverse deformations of the rim occur for longer duration, eventually leading to fingering (figure 7*b*–*f*).

The viscous time scales can also be estimated from the experimental measurements; for the sheet,

Lastly, the possibility of Kelvin–Helmholtz (K–H) type of instability at the interface of the jet is considered. The critical wavenumber *k*_{c} of the K–H instability for the air–water interface of a sheet with finite thickness under the long-wave approximation, derived from the dispersion relationship, is
*v*_{g}, *v*_{l} and *ρ*_{g} are the fluid velocities for gas and liquid, and the density for gas, respectively. Using *k*_{c}, the critical Weber number for the sheet can be calculated, *We*_{c}=2*πρ*_{l}Δ*v*^{2}/*γk*_{c}≈1500. Since the Weber number for the observed velocity and length scales in the present experiments is 56.2, we can safely ignore K–H instability in the following analysis.

## 4. Linear stability analysis

In this section, a linear stability analysis of the transverse disturbances in rim–sheet system is performed. Based on the discussion of the previous section, the fluid system will be treated as being incompressible, inviscid and irrotational with no tangential shear on the interface. We note that, by analogy with crown splashes [10,12], the capillary effect plays an important role in the axial deformation of the rim. In the flip-through case, there is an additional transverse instability mode triggered through the undulations in the sheet and this may be coupled to the rim instability, leading to break up of the rim–sheet system.

### (a) Governing equation

The model of the rim–sheet system, presented here, is based on mass and momentum conservation for the both of the rim and fluid sheet of incompressible inviscid fluid, which is derived as an extension of Agbaglah *et al.* [12]. While the axial stretching of the rim, due to the radial crown expansion, decreases the amplitude of the perturbation [13], no such assumption can be made here as the planar sheet is initially ejected from the two-dimensional wave crest without any initial transverse stretching (see §3f).

The longitudinal oscillation of the sheet thickness, whose amplitude decreases with the distance to the rim, has also been observed during retraction for low Ohnesorge number flows [30,31]. As yet the presence of any longitudinal oscillation has not been confirmed in the present experiments; however, the generation mechanism for the observed transverse variations in sheet thickness may be interpreted as a re-orientation of capillary waves on a thin stretched sheet [20]. As this may modify the instability, it should be included in the model of flip-through jets. Explicit mass and momentum exchanges between the rim and thinning sheet are also provided in the present model for describing the fully coupled dynamics of the rim and sheet system. Finally, since the length-scale characterizing the deformation process is very small gravitational effects can be neglected.

The edge of the rising planar sheet, of uniform thickness *h*(*x*,*t*), is bounded by a rim, with radius *a*(*x*,*t*), located at a vertical elevation of *η*(*x*,*t*). The horizontal and vertical fluid velocities of the rim (*u*_{r}(*x*,*t*), *v*_{r}(*x*,*t*)) and the film (*u*_{f}(*x*,*t*), *v*_{f}(*x*,*t*)) are defined as shown in figure 13. The equations describing the mass and momentum conservation, based on a long-wave approximation, are
*ρ* is the density of the liquid and *γ* is the surface tension. The last terms of the right-hand side of equations (4.2)–(4.4) present mass and momentum fluxes of liquid entering the rim from the sheet during the retraction [10].

Roisman *et al.* [10] and Agbaglar *et al.* [12], all assume constant sheet thickness and velocity in the previous studies for their work on the crown splash; this is an appropriate assumption in describing the rim dynamics for spatially uniform and steady state of the sheet. Roisman [14] introduced a constant velocity gradient on the undeformable free sheet to model the effect of steady stretching on the rim instability. These previous studies examined a simplified rim model assuming that the rim does not have an inverse effect on flow in the film [32]. However, in the flip-through event, the presence of free rims may affect flow inside the sheet resulting in thinning deformation (figure 11). In this case, the sheet dynamics should be solved simultaneously with the rim model (4.2)–(4.4) [32]. The mass and momentum conservation equations for a thin inviscid liquid sheet [32,19] are

In the flip-through case, variable transverse motion of a thinning sheet has been observed (figure 11); consequently, the sheet parameters *h*(*x*,*t*), *u*_{f}(*x*,*t*), *v*_{f}(*x*,*t*) are functions of time as well as space (*x*) assuming vertically uniform flow in the sheet. As the mass flux from the sheet to the rim in the retraction (right-hand side of equation (4.2)) decreases, mass in the entire sheet, integrating (4.5) from *y*=0 to *η*, the mass balance of the sheet yields
*y*=0 to *η*, represent the mechanical balance of inertia of the sheet, surface tension and the momentum flux transferred from the sheet to the rim

All the variables are non-dimensionalized with the reference rim radius *a*_{r} and capillary time *x**=*x*/*a*_{r}, *η**=*η*/*a*_{r}, *a**=*a*/*a*_{r}, *h**=*h*/*a*_{r},

### (b) First-order analysis

Initially, a spatially independent flow which is uniform in the cross-sheet, *x*, direction is considered. Such a system has a uniform rim of radius, *a*_{0}(*t*), rim location, *η*_{0}(*t*), and rise velocity, *v*_{r0}(*t*). It is attached to a uniform rising fluid sheet of thickness, *h*_{0}(*t*), with velocity, *v*_{f0}(*t*). Introducing small perturbations to the base state (figure 13*b*) gives *η*(*x*,*t*)=*η*_{0}(*t*)+*η*_{1}(*x*,*t*), *a*(*x*,*t*)=*a*_{0}(*t*)+*a*_{1}(*x*,*t*), *u*_{r}(*x*,*t*)=*u*_{r1}(*x*,*t*), *v*_{r}(*x*,*t*)=*v*_{r0}(*t*)+*v*_{r1}(*x*,*t*), *h*(*x*,*t*)=*h*_{0}(*t*)+*h*_{1}(*x*,*t*), *u*_{f}(*x*,*t*)=*u*_{f1}(*x*,*t*) and *v*_{f}(*x*,*t*)=*v*_{f0}(*t*)+*v*_{f1}(*x*,*t*). It should be noted that in this case, where there is no transverse stretching and the impact is driven by a laterally uniform wave impact [4], there are no transverse base velocities for the rim or the sheet.

Substituting these variables into equations (4.11)–(4.17), the equation system for the first-order perturbations is given by
*w*_{0}=*v*_{f0}−*v*_{r0}.

### (c) Base flow

In the current analysis, the base solutions for the zeroth-order variables at the initial state of the system are considered, assuming that base-flow changes much more slowly than the perturbations. Consequently, any temporal changes of the base state may be neglected. This, so-called, frozen approximation has often been used for linear stability analysis with time-variant base solutions [10,12]. In this case, equations (4.12) and (4.14) provide the relations of the initial rim radius *a*_{i}, rim velocity *v*_{ri} and sheet thickness *h*_{i}
*w*_{i} is the initial relative vertical velocity between *v*_{ri} and the sheet velocity *v*_{fi}. Using the above equations, a relationship between the temporal change of the initial radius, *a*_{i}, *h*_{i}, *η*_{i} and *a*_{i}, *h*_{i}, *η*_{i} which determine the base state of the system and which are used as parameters in this analysis. The equation system (4.18)–(4.24) may be linearized with constant coefficients determined from the frozen parameters; *a*_{0}≈*a*_{i}, *h*_{0}≈*h*_{i}, *η*_{0}≈*η*_{i}, *v*_{r0}≈*v*_{ri} and *v*_{f0}≈*v*_{fi}. It should be noted that the initial sheet length, *η*_{i}, is used as a parameter to describe the degree of the thinning effect owing to the stretching of the sheet rather than an exact measure of the initial sheet length.

### (d) Solutions for perturbations

Introducing perturbational of the from
*σ* is the growth rate and *k* is the wavenumber, into (4.15)–(4.21) and solving gives

Writing this system in matrix form, as
**M** can be expressed in terms of four parameters, *η*_{i}, *a*_{i}, *h*_{i} and *w*_{i} by using equations (4.27)–(4.29). The determinant of **M** must be zero to provide non-zero solutions
*σ*>0, and attenuate when *σ*<0. Multiple eigenvalues for an identical wavenumber may be obtained as solutions of the polynomial equation, depending on the initial frozen conditions of the system.

### (e) Mechanical effects of a thinning film

Here, we will discuss the effect of the thinning of the rising sheet on the dynamics of the rim–sheet system. Thinning drives the deformation process of the flip-through jets observed in figures 2 and 7, which is distinct from that observed in crown splashes.

The initial sheet length *η*_{i} can be considered as a parameter describing the degree of thinning, since the thickness of the whole sheet over length *η*_{i} is assumed to be uniformly reduced by the transfer of mass to the growing rim in the space independent base state. Thus, the thickness of a longer sheet undergoes less change for particular flux.

Assuming an infinitely long sheet, *η*_{i}≫*h*_{0}, *u*_{f}′, *v*_{f}′ and *h*′ become negligible in the case where *ση*_{i}≫*w*_{i} for real positive *σ*, under these circumstances equations (4.31)–(4.34) reduce to the retraction-based rim instability model of Agbaglah *et al.* [12].

In the special case of no momentum exchange between the rim and sheet, *w*_{i}=0 (or *v*_{f1}, equations (4.45) and (4.46) represent the dynamic beam equations
*ω*^{2}=*h*_{i}*k*^{4} for periodically oscillating capillary waves in *x*-direction. This solution corresponds to symmetrical waves on a thin sheet at the long-wave limit presented by Taylor [15], and that amplification of this type of capillary waves on a thinning sheet is described by Longuet-Higgins [20] (see also figure 13). Since there is no growth of the perturbation in this closed equation system for the sheet, no capillary contribution is provided to the rim dynamics in this case of no sheet stretch.

The eigenvalue equation for (4.31)–(4.34) at this limit shows an RT instability mode will be present, as follows:

In another limit *h*′ and the perturbed rim location *η*′ can be related
*et al.* [33] introduced the local velocity gradient in the sheet ∂*v*_{fi}/∂*y* as a parameter describing rate of stretching into the model of the rim behaviour. In the current model, the gradient of sheet velocity is modelled as *et al.* [33] except the viscous shear force which is ignored in the current model.

In the present experimental results, we find transitional sheet dynamics, where a thick sheet is ejected from the standing wave crest at high acceleration leading to rapid thinning as the sheet elongates longitudinally (figure 11). In this case, the transitional sheet lengths are modelled by the finite, moderate, range of *η*_{i} presenting a case which has not been analysed previously. For finite moderate sheet lengths (with both thinning and capillary effects), more complex mass and momentum exchanges between the rim and sheet induce multiple instability modes in the rim–sheet system.

### (f) Results

The unstable modes of the coupled rim–sheet system are first compared with the rim instability of a crown splash. This comparison aims to identify the contributions of the sheet dynamics to the behaviour of the splashing jet system.

As already discussed, the current model with no thinning effect (i.e. *η*_{i}=1.0×10^{7}, are completely consistent with the previous rim model for any initial rim conditions (figure 14). The models provide the fundamental feature of the rim stability; growth rate with the most unstable wavenumber at *k*∼0.7, and destabilization owing to decrease of *h*_{i} and *k* (equation (4.51)) and thus follows the RP mechanism in case *a*). In the case of a thin sheet with negative acceleration *b*), the RT mode defines a steeper gradient of the growth rate at the lowest wavenumber (as

Multiple instability modes emerge in competing mechanisms between capillary, retraction and sheet stretch mechanisms of the rim–sheet system, which are sensitive to the initial sheet thickness and rim acceleration. The largest and second largest eigenvalues for the case of *η*_{i}=10 and *h*_{i}=0.1 are shown in figure 15. While the wavenumber achieving the maximum growth rate (*k*∼0.7) is identical with those in figure 14, the unstable range extends into the low wavenumbers. Roisman *et al.* [33] introduced the velocity gradient in the sheet, *S*=∂*v*_{fi}/∂*y* as a model of stretch effect to the rim stability of a crown splash. They found that destabilization of the rim at low wavenumbers in 0.1≤*S*≤5.0, which is related to the weakening of the capillary effects, is less pronounced for long waves. As *et al.* [33] (see §4e). In figure 15, the velocity gradient *S*≈*w*_{i}/*η*_{i} lies in the range 0.32 to 0.54. This is within the range discussed by Roisman indicating that an identical mechanism may be responsible for destabilizing the long-wave motions. This can be readily verified from the eigenvalues, *σ*_{s}, for equations (4.56)–(4.58) at *k*=0 for the simplest case of *σ*_{s} takes real positive value when the sheet thinning parameters *b*), the second eigenvalues increases becoming positive, this results in two maximal growth rates at *k*∼0.7 for the first eigenvalue and in low wavenumbers of *k*<0.5 for the second one. This is relevant to the observed double spectrum peaks for the surface deformation (figure 10).

We will now consider the relative importance of the capillary and stretch effects during the retraction process on the stability of the rim–sheet system. While the long-wave assumption is not valid for shorter scale fluctuations, *k*>1, the growth rates of features of the rim and sheet associated with the largest eigenvalues (figure 16) may still be interpreted. For the case of high thinning rate of the sheet (*η*_{i}=1.0), the systems with small rims, *a*_{i}≤0.1, are unstable over the whole range of wavenumbers. Longuet-Higgins [20] showed the growth amplitude (*ζ*′) of perturbed surface displacements for symmetric long waves on a thin sheet, with thickness 2*h*, stretched in the direction perpendicular to wave propagation on the basis of conservation of wave action is *ζ*′/*h*∝*h*^{−1/4}, indicating the relative amplitude grows on the stretched sheet regardless of wavenumber. We have also confirmed larger growth rates for thinner sheets at high rate of stretch. The sheet instability at the lowest wavenumber (amplifications of laterally uniform perturbations) is owing to unsteady effect of the thinning sheet as discussed above, which may result in rapid decrease of the perturbed sheet thickness along the neck of the lateral rim at the initial stage of flip-through in figure 7*a*–*c* (see also figure 2). The increase of rim size *a*_{i} contributes to stabilize the system especially in high wavenumbers, since the thinning effect relatively weakens (positive *σ*_{s} decreases with *a*_{i}) and thus the capillary effect accounts for the decay of the amplification rate in the regime of high wavenumbers. As the sheet stretching attenuates, the systems for any rim size are stabilized over the entire wavenumbers especially in the lowest wavenumbers (figure 16*b*,*c*).

Figure 17 shows the comparisons of the growth rates as functions of the sheet thickness, thinning rate and initial rim acceleration. In the case of high thinning rate, *η*_{i}=1.0, the stretching effect on the sheet governs the stability of the system, resulting in significant growth rates over the entire long-wave regime, 0≤*k*≤1, for (*a*) *k*∼0.5 in (*a*) *τ*_{ch} and rim *τ*_{cr}, experimentally estimated in §3f, may be relevant to the multiple instability growth modes. In (*a*), *k*∼0.6 regardless of the sheet thickness, which also indicates the two instability modes simultaneously occurring in 0<*k*<0.6, while no instability amplification is observed in *k*>0.6. With increase of *η*_{i} for *k*∼0.7 (see (*b*,*c*) *k*∼0.7 for the neutral rim acceleration with *η*_{i}=5.0 and 10.0 (see (*b*,*c*) *h*_{i} for any cases, which may result in fundamental differences in the deformation of a thicker jet formed at pre-breaking impact (figure 8*a*,*b*) from the flip-through case (figure 7). Figure 17*d* *b*), that is RP and RT mechanisms destabilize the perturbed rim without available stretch effect. In positive rim acceleration, *η*_{i}). The wavenumber selections for these multiple instability modes, depending on transitions of the rim acceleration, sheet thickness and rim size, may occur during the flip-through event.

## 5. Mechanisms of transverse deformation of the flip-through jets

Although the frozen approximation used in the stability analysis cannot provide fully time-dependent stabilities throughout the event, the assumption of much faster growth of perturbations than the base state deformation of the jet still allows the use of the growth rates for analytical parameters corresponding to the experimental ones at sequential stages of the flip-through (as shown in figure 7). Figure 18 shows the observed spectra for the transverse deformation of the flip-through jets at the phase achieving the maximum value and the vertical distributions of the observed rim acceleration *a*_{e} and capillary time *τ*_{cr}, see §3f). The wavenumbers for the observed spectrum are explicitly compared with the ones for the analytical instability growth rates for the corresponding *b*, the initial rim acceleration in the case of flip-through (*x*=0.0 mm) decreases from *z*=100 mm to *z*=200 mm owing to drag force and gravity. Therefore, the rim accelerations, *x*=0.0 mm) of figure 18*a*, a single spectral peak initially appears at *k*∼0.5 (red broken line at *z*=100 mm), which subsequently evolves into a wider width spectrum. We find that the peak position, *k*_{p1}, moves to lower wavenumbers with the measurement level *z*; *k*_{p1}∼0.5 at *z*=100 mm, 0.4 at *z*=150 mm and 0.35 at *z*=200 mm (see red arrows in figure 18*a*), and that the peak width extends towards lower wavenumbers. Another spectrum peak then emerges at *k*_{p2}∼0.7 at *z*=200 mm, forming double peak spectrum (red solid line). In the quasi-flip-through, the identical wavenumber shift (*k*_{p1}∼ 0.35–0.3, see black arrows) and the width extension for the lower wavenumber peak are observed while the peak position for the higher one (*k*_{p2}∼0.7) is unchanged. These transitional features of the spectrum are relevant to the wavenumber selections during the transitional base state as discussed bellow.

In the flip-through, at the level *z*≤100 mm of figure 18*b*, the strong impact pressures rapidly accelerate the up rushing planar sheet. Since the stretching, and consequent thinning, of the sheet with positive acceleration (*a*) *k*≤0.6 may be associated with the observed prominent spectrum in *k*∼0.4−0.6 at *z*=100 mm. It should be noted that the observed spectrum for the integrate length only reflects vertical displacement of the edges of jets not thickness (see §3d), while the analytical instability at the lowest numbers is relevant to long-scale variations of the sheet thickness. This may cause discrepancies between the observed spectrum and the analytical growth rates in low wavenumbers. As the rim acceleration then decreases to zero at *z*=150 mm before achieving *z*=200 mm (figure 18*b*), the sheet stretch attenuates in the transitional state corresponding to (*a*)–(*c*) *η*_{i}, the capillary instability with the growth at *k*∼0.7 becomes the dominant mechanism for the unstable rim–sheet behaviour, since the maximum growth rates at the lowest wavenumber decay as do the second eigenvalues in the middle range. We also find the wavenumber shift for the lower range of the maximal growth rates as the sheet stretch attenuates (*η*_{i} increases); the unstable range of *k*<0.65 for *η*_{i}=1.0, *k*<0.3 for *η*_{i}=5.0 and *k*<0.2 for *η*_{i}=10.0. The variations of the analytical unstable ranges suggest the attenuation of the sheet stretch in the decelerating jets can attain the wavenumber shift of the spectrum peak, *k*_{p1}, and the extension of the peak width towards low wavenumbers as the jet rises. Finally, the RP and RT instabilities govern the perturbation growth in the retraction process under less effect of the sheet stretch (see figure 17 (*c*) or (*d*) *k*∼0.7 corresponding to the higher one of the peak wavenumbers (*k*_{p2}) observed at *z*=200 mm in figure 18.

Although the current analysis does not explain large surface displacements beyond linear perturbations, the wavenumber selections for the instability modes responding to sequential base flow states consistently interpret a series of the deformations of the rim and the sheet at the early stage in figure 7. Laterally, uniform thinning on the sheet along the neck of the rim first emerges together with long-scale variations of the sheet thickness at about 1–2 cm intervals (stripe patterns of the image brightness) owing to the sheet stretch at phases (*a*) and (*b*), which corresponds to the initial instability growth in low wavenumbers (figure 17*a* *c*) and (*d*), owing to the wavenumber shift. The fingers are formed at the intervals of less than 1 cm, corresponding to the maximum capillary instability growth at *k*∼0.7, on the rim vertically undulating in synchronization with the long-scale sheet deformations.

On the other hand, there are apparent differences in the dominant wavenumbers for pre-breaking case (blue line in figure 18) and those in the flip-through event. This may be caused by the modification of unstable behaviours in the relatively thick, initial, jet (which limits capillary effects) and also through viscous effects at the wall contacting the liquid sheet. This maybe related to fingering instability of a spreading film [25].

## 6. Conclusion

The unstable behaviour and transverse deformation of violent up rushing fluid jets generated by a flip-trough event occurring when a steep breaking wave impacts on a vertical wall have been characterized using image analysis. Regular waves generated in a flume, shoal and then break on to a backlit vertical wall, where the up rushing jet is captured with a high-speed video camera.

Analysis of the video images shows that the up rushing jet is formed of a fluid sheet with a free rim at the leading edge, similar to that observed in the well-known crown splash. The rim undulates about the transverse axis to form organized regular cusps which are amplified and eventually form finger jets.

On the other hand, we have found inherent behaviours of the rim bounded sheet that are distinct from crown splashes. As the sheet extends from the standing wave crest, it undergoes rapid stretching and consequently thinning, forming transverse undulations in sheet thickness. These undulations in the sheet are amplified by stretching and eventually rupture the sheet to form regular vertical ligaments. The finger shaped rim sections connected to these ligaments become detached from the main jet and break up to form spray. Computing the wavenumber spectra of the transverse displacements of the flip-through jet shows two dominant wavenumbers.

A linear stability analysis of a model frozen, coupled, rim–sheet system has been performed. In this analysis, the thinning sheet is modelled using a sheet length parameter to represent the stretching of the accelerating fluid sheet. A comparison of growth rates with respect to wave numbers between the present model (with no stretching) and the results presented by Agbaglah *et al*. [12], for rim instability under capillary and retraction dynamics, has been used to validate the model.

The relative mechanical roles played by the stretching and capillary effects acting on the rim on the stability of the fluid sheet are discussed in terms of linear perturbations. A system with a small rim and a highly stretched sheet is shown to be unstable over the entire long-wave range. According to Longuet-Higgins [20], the relative amplitude of symmetric capillary waves with respect to thickness of a thin sheet (without a rim) stretched in the direction perpendicular to propagation is proportional to

Mechanisms to induce sequential transverse deformation of the rim–sheet system, leading to the double peak spectrum form of the surface displacement, are discussed in terms of instability growth defined by the analytical parameters corresponding to the experimental ones at sequential stages of the flip-through.

In the initial stage of flip-through, where sheet dynamics governs the stability of the rim–sheet system, two comparable positive eigenvalues appear in the low wavenumber range resulting in large-scale variations in sheet thickness along the lateral rim. The presence of real positive first and second eigenvalues indicates instabilities grow with two distinct time-scales at a given wavenumber. Therefore, these perturbations are amplified at different velocities defined by the instability modes. As the rim decelerates owing to drag force and gravity and thus the stretch attenuates, two maxima in growth rate are observed at the lowest wavenumbers and at *k*∼0.7. The low wavenumber range containing the growth rate maxima decreases with attenuation of the sheet stretch. This wavenumber shift in the unstable range has a role to change in wavelength of the sheet thickness and long-scale vertical displacement of the rim location. Finally, the capillary instability with the most unstable wavenumber of *k*∼0.7 governs the system on the less stretch sheet with negative rim acceleration, which enhances short-scale rim deformations leading to finger formation. The sequential variations of the instantaneous instability growths for these multiple instability modes, depending on the sheet stretch, thickness, rim size and the accelerations, determines the integral features of the displacements of the ascending flip-through jets, resulting in the observed double peak spectrum.

## Data accessibility

The experimental image data supporting this article have been uploaded as the electronic supplementary material.

## Authors' contributions

Y.W. and D.I. conceived of the research, drafted the manuscript and approved the study for publication. Y.W. designed the study and carried out the analyses.

## Competing interests

The authors declare that no competing interests exist.

## Funding

This research was supported by JSPS Grant-in-Aid for Scientific Research (15H04043).

## Acknowledgements

The authors thank S. Ishizaki for his help in conducting experiments.

- Received June 15, 2015.
- Accepted September 18, 2015.

- © 2015 The Author(s)