4.2.1 Class I (flip-through) impact – analysis of pressure measurements

In the flip-through cases, the impact pressure on the wall is found to be very sensitive to the detailed geometry of the water surface just before the moment of impact, as is demonstrated below. Here we consider the $z_{b}=-0.113\unicode[STIX]{x1D706}_{0}$ case, which has been discussed above using photographs from a high-speed movie in figure 5 and the corresponding water surface profile history and pressure contour plots in figures 6(a) and 16(a), respectively. In figure 16(a), a double pressure peak is found. In order to understand the double peak phenomenon, LIF movies with higher temporal resolution and zoomed in at the zone of impact were taken. Four images from one of these movies are shown in figure 22. The surface profiles from the movie and the pressure measurements are shown in figures 23 and 24, respectively. The instants when the images in 22(a), 22(b), 22(c) and 22(d) are taken are called $t_{a}$ , $t_{b}$ , $t_{c}$ and $t_{d}$ , respectively, and are defined in the caption of figure 22 and marked on the plots in figure 24(a,b), see figure captions for additional details.

The double pressure peak can be seen clearly in figure 24(b). The magnitudes of the two pressure peaks are significantly larger than the pressures measured at the other heights over all time. The earlier peak and later peak were measured by two adjacent pressure sensors, 19 and 20, respectively. The time $t_{a}$ is the instant when the earlier pressure peak starts to rise from zero, according to figure 24(b). Figure 22(a) shows that at $t=t_{a}$ , the contact point has climbed slightly above the lower boundary of sensor 19. At $t=t_{b}$ , the earlier peak reaches its maximum value, see figure 24(b). As shown in figure 22(b), at $t=t_{b}$ , the contact point almost reaches the upper boundary of the sensor 19. At this instant, the tip of a small jet formed at the crest, see figure 22(b), is still upstream of the wall. This indicates there has not yet been direct impact of the jet with the wall. Therefore, the earlier pressure peak is probably caused by the focusing of the surface. The later peak has a higher maximum than the earlier peak and is measured by pressure sensor 20. As shown in figure 22(c), at $t=t_{c}$ , when the later peak starts to rise, the small jet on the surface just impinges on pressure sensor 20. This indicates that the later peak (with the highest magnitude) might be caused by the direct impact of a small jet on the wall. The later peak reaches its maximum at $t=t_{d}$ when the small jet has already impinged on the wall, see figure 22(d). These synchronized images and pressure records provide strong evidence that the earlier pressure peak on the wall is a unique feature of flip-through impact, as described in some previous theoretical work, and the higher second peak is likely to have been strongly influenced by the impact of the jet. However, the finite size and slight separation of the two pressure sensors, indicates that pressure measurements with improved spatial resolution would be needed to prove this hypothesis.

Figure 23.

Surface profiles near the impact zone before the moment of impact for condition $z_{b}=-0.113\unicode[STIX]{x1D706}_{0}$ . These profiles were obtained from the movie corresponding to figure 22. The profiles are from every other movie frame and the time interval between adjacent profiles is 0.149 ms ( $=0.000171/f_{0}$ ). The total time range for these profiles is 13.33 ms ( $=0.01533/f_{0}$ ). The profiles shown in bold correspond to the images shown in figure 22(a–d).

Figure 24.

A contour plot for the pressure evolution on the wall at $z_{b}=-0.113\unicode[STIX]{x1D706}_{0}$ is shown in (a). This is a zoomed in view of the region of maximum pressure from figure 16(a). The horizontal lines represent the centrelines of the pressure sensors; the sensor number increases monotonically with $z$ and sensors 19 and 20 are marked on the figure. The time histories of the pressures measured by sensors 14 to 20 are shown in (b); the highest pressure is measured by sensor 20. The four vertical white lines in (a) and grey dotted lines in (b) from left to right represent the times corresponding to the images shown in  22(a), 22(b), 22(c) and 22(d), respectively.

4.2.2 Class I (flip-through) impact – theoretical analysis

With existing theory, it is possible to understand better the physics that creates these temporally and spatially localized high-pressure regions during flip-through impact. Since the time scale of the flip-through impact is short, viscous effects from the wall cannot affect the outer flow and the phenomenon can be well described by potential flow theory. According to Bernoulli’s equation, the pressure on the wall consists of two parts: the dynamic pressure (defined herein as the sum of the unsteady term $\unicode[STIX]{x1D70C}\unicode[STIX]{x2202}\unicode[STIX]{x1D719}/\unicode[STIX]{x2202}t$ and the nonlinear term $0.5\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D735}\unicode[STIX]{x1D719})^{2}$ ) and the hydrostatic pressure ( $\unicode[STIX]{x1D70C}gz$ ). The potential flow model proposed by Cooker (Reference Cooker2002) describes several types of focusing flows with indented free surfaces, similar to that found in the present experiments. In one of the calculations presented in the paper, at any instant in time, the water surface consists of an horizontal portion and a semi-circular indentation with radius $R$ . It is assumed the flow field is described by a line sink placed at the centre of the semi-circle. The instantaneous normal velocity of any point on the circular arc is $V$ and the normal velocity on the surrounding flat free surface is zero. Given $R$ and $V$ at any instant in time, the theory predicts the pressure field in the fluid domain $r\geqslant R$ as

(4.1) $$\begin{eqnarray}\displaystyle \displaystyle \frac{p(r,\unicode[STIX]{x1D703})}{\unicode[STIX]{x1D70C}V^{2}} & = & \displaystyle \frac{1}{Fr^{2}}\cos \unicode[STIX]{x1D703}\left(\frac{r}{R}\right)+\left(\frac{8}{3\unicode[STIX]{x03C0}}-\frac{1}{Fr^{2}}\right)\cos \unicode[STIX]{x1D703}\left(\frac{R}{r}\right)\nonumber\\ \displaystyle & & \displaystyle -\,\frac{1}{2}(1+\cos 2\unicode[STIX]{x1D703})\left(\frac{R}{r}\right)^{2}-\frac{1}{2}\mathop{\sum }_{n=2}^{\infty }a_{n}\cos [2n-1]\unicode[STIX]{x1D703}\left(\frac{R}{r}\right)^{2n-1},\end{eqnarray}$$

where the plane polar angle $\unicode[STIX]{x1D703}$ is measured from the vertically downward direction and

(4.2) $$\begin{eqnarray}\displaystyle & \displaystyle a_{n}=\frac{16}{\unicode[STIX]{x03C0}}\frac{(-1)^{n+1}}{(4n^{2}-1)(2n-3)}, & \displaystyle\end{eqnarray}$$

and $Fr$ is the Froude number defined as $Fr=V/\sqrt{gR}$ . On the line of symmetry at $\unicode[STIX]{x1D703}=0^{\circ }$ (the front face of the structure in the present experiments), equation (4.1) reduces to

(4.3) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{p_{w}(r)}{\unicode[STIX]{x1D70C}V^{2}}=\frac{1}{Fr^{2}}\left(\frac{r}{R}\right)+\left(\frac{8}{3\unicode[STIX]{x03C0}}-\frac{1}{Fr^{2}}\right)\left(\frac{R}{r}\right)-\left(\frac{R}{r}\right)^{2}-\frac{1}{2}\mathop{\sum }_{n=2}^{\infty }a_{n}\left(\frac{R}{r}\right)^{2n-1}. & \displaystyle\end{eqnarray}$$

The first term in (4.1) (and (4.3)) is multiplied by $1/Fr^{2}$ . This term is the hydrostatic pressure, which, of course, produces horizontal contours of constant pressure that increase linearly with depth. Only one other term in (4.1) and (4.3) is influenced by gravity, i.e. is multiplied by $1/Fr^{2}$ . This second term is part of the effect of unsteadiness, as explained in Cooker (Reference Cooker2002), and produces a pressure field that is negative in the domain of the fluid and decreases in magnitude monotonically with increasing $r/R$ at fixed $\unicode[STIX]{x1D703}$ . Herein, the combined contribution from these two terms is called the gravitational effect, and it is found numerically that this part of the pressure increases monotonically with increasing depth at any horizontal position. The pressure from the combined effect of the remaining terms (those independent of $Fr$ ) is herein called the inertial effect and is plotted in figure 2 in Cooker (Reference Cooker2002). These contours of dimensionless pressure are only a function of $R/r$ and $\unicode[STIX]{x1D703}$ and include a subsurface pressure maximum. Thus, whether or not a subsurface maximum exists in the total pressure field depends on the instantaneous value of $Fr$ since it controls the relative magnitude of the gravitational and inertial pressure fields. In the unsteady situation of the flip-through wave impact of the present experiments, the instantaneous value of $Fr$ increases continuously with time, as will be demonstrated below. As the impact begins, $Fr$ is small and the gravitational effect dominates. As a result, equation (4.1) indicates that no pressure maximum exists below the free surface, as shown in figure 25(a) ( $Fr=5.78$ ). After $Fr$ reaches a critical value ( $Fr_{m}\approx 7$ as predicted by (4.3)), the inertial effect becomes dominant and a localized maximum pressure appears, as is shown in figure 25(b) ( $Fr=9.80$ ). Thus, prediction of the time when the critical Froude number is reached and the subsurface pressure maximum occurs in the experiment requires the function $Fr(t)$ . The determination of this function from the experimental data is addressed in the following.

Figure 25.

Pressure distributions in the flow field computed by (4.1) at $Fr=5.78$ and $Fr=9.80$ are shown in (a) and (b), respectively. Temporal evolution of the pressure distribution on the wall computed from (4.3) is shown in (c). The two vertical dashed lines on the left and right in (c) represent the times corresponding to the pressure fields in (a) and (b), respectively. In this calculation, the required temporal variation of the contact point height and Froude number is taken from the experimental data, see text for details.

Though Cooker’s model is not directly applicable to the present experiments, it can be used to gain some insight into the flow dynamics. The difference between the two flows includes the boundary shape outside the circular arc and the distribution of normal velocity along the circular arc. Thus, in order to apply the model to the experiments, we need to make some choices forced by the differences in the surface shapes and motions. An alternative choice for the radius in the experiments is the length scale $L(t)$ defined as the height difference between the contact point and the crest. Since the averaged surface normal velocity on the circular arc is difficult to obtain, particularly when the surface shrinks to a small size, the contact point velocity $V_{cp}$ will be chosen as the velocity scale. Plots of $L$ and $V_{cp}$ versus time are given in figure 26 for one experimental run at $z_{b}=-0.113\unicode[STIX]{x1D706}_{0}$ . The solid curve in each plot is a least-squares fit of a power-law function to the data, see figure caption for equations.

Figure 26.

The temporal evolution of length scale $L$ and velocity scale $V_{cp}$ from the experiment are shown in (a) and (b), respectively. See text for the definition of $L$ and $V_{cp}$ . In (b), the circles are the velocity values computed from finite difference of the contact point position versus time data from the experiments. The solid curves in (a) and (b) are power-law functions that were obtained by least-squares fits to the experimental data: (a)  $L/\unicode[STIX]{x1D706}_{0}=0.1366|tf_{0}|^{0.637}$ and (b)  $V_{cp}/\bar{c_{p}}=0.3214|tf_{0}|^{-0.304}$ . The proximity of the exponents $0.637$ and $-0.304$ to $2/3$ and $-1/3$ , respectively, is discussed in § 4.2.4.

Using the data from figure 26, the evolution of $Fr$ versus time before the moment of impact is calculated and plotted in figure 27(a). The spread of data points mostly comes from the numerical error when taking derivatives of the contact point position to obtain $V_{cp}(t)$ . Starting from $tf_{0}=-0.015$ , $Fr$ gradually increases following a nearly linear curve. After approximately $tf_{0}=-0.008$ , $Fr$ increases rapidly over time. From the pressure record shown in figure 24, when $tf_{0}>-0.00225$ (shown by the first dash vertical line), the pressure measured by the sensor just below the free surface is larger than the pressures measured by all the lower sensors. This indicates the pressure maximum starts to occur after this instant. In figure 27(a), this time is labelled as $t_{e}f_{0}$ and the corresponding value of $Fr$ is $Fr_{e}\approx 15$ . At later times, $Fr$ exceeds $Fr_{e}$ and a pressure maximum always occurs below the contact point.

Cooker’s theory can also be used to create a theoretical/experimental pressure contour plot for direct comparison with the plot in figure 24(a). In addition to $Fr(t)$ and $L(t)$ , the calculation requires the contact point height data, $z_{cp}(t)$ , which was also used to determine $L(t)$ . The resulting pressure contour plot is given in figure 25(c). This prediction is qualitatively similar to the experimentally measured data shown in the other figure; however, the spatially localized pressure maximum, appears after approximately $tf_{0}=-0.0088$ , corresponding to $Fr\approx 7$ . Given the differences between the flows in experiments and theory, the mismatch of the critical Froude numbers and times is to be expected. However, both the experimental results and Cooker’s model strongly support the idea that the instantaneous $Fr$ determines whether a pressure maximum exists below the free surface.

Figure 27.

Froude number ( $Fr$ ) and Weber number ( $We$ ) versus time from experiments are presented in (a) and (b), respectively. The length scale and velocity scale used to compute $Fr$ and $We$ are taken from the data shown in figure 26. The vertical lines in (a) and (b) represent the time when the local pressure maximum starts to occur in the experiments. The inset of (a) is the close up view of the main plot. The solid curve in (a) is the function $Fr(t)=0.367|tf_{0}|^{-0.623}$ , obtained from the power-law fitting of $L(t)$ and $V_{cp}(t)$ shown in figure 26.

Similar features of the impact pressure in the flip-through condition are also reported in Bredmose et al. (Reference Bredmose, Peregrine and Bullock2009). In this work, numerical simulations of a shallow-water wave impact on a wall that extends to the bottom of the tank are performed. In the flip-through condition, their results (figure 12 in the paper) show an isolated pressure maximum on the wall approximately 10 ms ( $=0.0115/f_{0}$ ) before the impact, analogous to the present experiments. At early times before impact, the pressure on the wall increases with depth. The pressure peak only lasts for approximately 15 ms ( $=0.0173/f_{0}$ ).

4.2.3 Class I (flip-through) impact – the vertical jet

The acceleration of the high-speed vertical jet has received considerable attention in the literature, see for example Cooker & Peregrine (Reference Cooker, Peregrine, Banner and Grimshaw1992). Previous studies have shown that this high acceleration is caused by the high pressure gradient between the location of the maximum pressure and the free surface. However, it is difficult to measure the acceleration of the flow directly. Instead, the acceleration of the flow near the contact point can be estimated by twice differentiating the contact point position versus time data (measured at a frame rate of 13.5 kHz). To avoid the noise created by a second-order finite difference method on the position versus time data, a third-order polynomial fitting function is applied to the contact point position versus time data and the acceleration of the contact point is estimated from the second-order derivative of this fitting function. Using this method, it is found that the estimated acceleration of the contact point can reach as high as $481g$ in the present experiments. Due to the limitations of the temporal resolution and the accuracy of the curve fitting, the exact value of the maximum contact point acceleration is still difficult to obtain. Nevertheless, the high estimated value provides additional evidence of the existence of a subsurface region with a high pressure gradient.

4.2.4 Class I (flip-through) impact – surface tension

As the moment of impact is approached, the growth of the small-scale features on the water surface profile (see the profile history in figure 23) may indicate that surface tension begins to play an important role. To explore this possibility, the Weber number at each time,

(4.4) $$\begin{eqnarray}\displaystyle & \displaystyle We(t)=\frac{\unicode[STIX]{x1D70C}V_{cp}(t)^{2}L(t)}{\unicode[STIX]{x1D70E}}, & \displaystyle\end{eqnarray}$$

was calculated from the length and velocity scales plotted in figure 26 and the result, $We$ versus $t$ , is plotted in figure 27(b). The values of $We$ are spread around a constant level approximately $We=400$ for most of the impact process, indicating that inertia effects dominate surface tension. This is consistent with power-law scaling functions proportional to $t^{1/3}$ and $t^{-2/3}$ for $V_{cp}$ and $L$ , respectively, see the caption to figure 26 and the discussion in, for example, Longuet-Higgins (Reference Longuet-Higgins1993). Near the moment of impact, $-0.0025<tf_{0}<0$ , $We$ decreases from the constant level to nearly zero. This latter behaviour is consistent with $V_{cp}$ tending to a finite value and $L$ tending to zero as $t$ tends to $0$ . Using the power-law functions for $L(t)$ and $V_{cp}(t)$ as given in the caption of figure 26, or the corresponding rational values of $2/3$ and $-1/3$ , to represent $L(t)$ and $V_{cp}(t)$ in the calculation of $We(t)$ does not produce the final stage of the data in figure 27(b) because the power laws do not fit the $L$ - $V_{cp}$ data well at small times. In particular, the $V_{cp}(t)$ power law tends to $\infty$ as $t$ tends to zero, rather than a maximum value as observed in the experiments. This late time inconsistency between the power-law functions and the data is as expected for a real physical system and may indicate an ultraviolet cutoff of the divergent process, as is also described in Zeff et al. (Reference Zeff, Kleber, Fineberg and Lathrop2000).

4.2.5 Class II impact – pressure oscillations

One of the very prominent features of the pressure contour plot at $z_{b}=0.0$ , shown in figure 16(b), is the pressure oscillation that occurs nearly in phase over the vertical row of sensors. Images in figure 9 indicate that, during the impact process, a crater exists between the front face of the structure and the growing jet that issues from the crest. When the jet meets the rising water column, it is possible that part of the air within the crater cannot escape. Therefore, it is very likely that the pressure oscillation is caused by the entrainment of air bubbles. Unfortunately, the LIF movies and other visualizations that have been attempted are not able to see into this flow region to observe the bubbles. However, bubble oscillation theory will be used here to compute the diameter of a spherical bubble with the oscillation frequency measured in the experiments. The natural frequency of a spherical bubble in water can be estimated by (Plesset & Prosperetti Reference Plesset and Prosperetti1977)

(4.5) $$\begin{eqnarray}\displaystyle & \displaystyle f_{n}=\frac{1}{2\unicode[STIX]{x03C0}R_{0}}\sqrt{\frac{3kP_{eq}}{\unicode[STIX]{x1D70C}}-\frac{2\unicode[STIX]{x1D70E}}{\unicode[STIX]{x1D70C}R_{0}}}, & \displaystyle\end{eqnarray}$$

where $R_{0}$ is the radius of the bubble at equilibrium state, $\unicode[STIX]{x1D70C}$ is the density of water, $k=1.4$ is the heat capacity ratio of air, $\unicode[STIX]{x1D70E}=73~\text{mN}~\text{m}^{-1}$ is the surface tension of water, $P_{eq}$ is the pressure inside the bubble at equilibrium state and

(4.6) $$\begin{eqnarray}\displaystyle & \displaystyle P_{eq}=P_{\infty }+\frac{2\unicode[STIX]{x1D70E}}{R_{0}}. & \displaystyle\end{eqnarray}$$

According to (4.5), the 1000 Hz pressure oscillation frequency corresponds to the natural frequency of a spherical air bubble with a radius of 3.28 mm under ambient pressure of 1 atm. At a later time, oscillations with higher frequencies (2800 Hz) are observed, corresponding to bubbles with radius as small as 1.17 mm. The low-frequency component of the oscillation at 1000 Hz covers a wide range of heights below the free surface with very little phase shift with increasing depth. This indicates that the acoustic waves that are most likely generated near the free surface transmit through the fluid at very high speed. In the computational work by Bredmose et al. (Reference Bredmose, Peregrine and Bullock2009, Reference Bredmose, Bullock and Hogg2015), in the case of air entrainment, similar pressure oscillations are found. In their simulation, in cases when the fluid has non-zero initial aeration level, the propagation speed of the acoustic waves is decreased, causing a more noticeable phase shift in the pressure oscillation with increasing depth.

4.2.6 Class III impact – analysis of pressure measurements

In the two cases of class III impact, two prominent differences are observed between the two conditions. First, the impact pressure measured at $z_{b}=0.043\unicode[STIX]{x1D706}_{0}$ is much higher than that measured at $z_{b}=0.022\unicode[STIX]{x1D706}_{0}$ . Comparison of the images in figures 13 and 14 shows that the rising water column along the wall at $z_{b}=0.043\unicode[STIX]{x1D706}_{0}$ is thinner than that at $z_{b}=0.022\unicode[STIX]{x1D706}_{0}$ . This difference may be caused by the smaller time available for the water column to develop before the plunging jet arrives in the $z_{b}=0.043\unicode[STIX]{x1D706}_{0}$ case. Perhaps then, for $z_{b}=0.043\unicode[STIX]{x1D706}_{0}$ , the horizontal momentum carried by the plunging jet is less attenuated by the water column during impact, causing the higher pressure at this condition. Second, a pressure oscillation occurs at $z_{b}=0.043\unicode[STIX]{x1D706}_{0}$ and not at $z_{b}=0.022\unicode[STIX]{x1D706}_{0}$ . This difference is likely to be associated with the differences in the detailed geometry of the water surface (see figures 13 and 14) after it first touches the bottom of the structure. It is hypothesized that air is entrained at $z_{b}=0.043\unicode[STIX]{x1D706}_{0}$ as a result of the collision between the plunging jet and a micro-jet (see figure 14 e), which is generated at the bottom corner of the structure at the initial impact. At the initial impact for the $z_{b}=0.022\unicode[STIX]{x1D706}_{0}$ case, a ripple is formed instead. This growing ripple eventually merges with the plunging jet in a smooth process without direct collision. The fluid momentum near the bottom corner of the structure at initial impact is probably relevant to these detailed responses of the water surface. To further examine this possibility, the temporal evolution of the vertical velocity ( $V_{s}$ ) of the water surface of the undisturbed breaker at the streamwise tank position of the vertical wall, $x=0$ , is shown in figure 28. The instants when the water surface at $x=0$ reaches the bottom corner of the structure (initial impact) for these two conditions are shown in the figure. The vertical velocity of the water surface at $x=0$ is $V_{s}=0.28\bar{c_{p}}$ at $z_{b}=0.022\unicode[STIX]{x1D706}_{0}$ and $V_{s}=0.73\bar{c_{p}}$ at $z_{b}=0.043\unicode[STIX]{x1D706}_{0}$ . The larger $V_{s}$ indicates the water surface impacts the bottom of the structure with more momentum. This may contribute to the formation of the micro-jet in the case $z_{b}=0.043\unicode[STIX]{x1D706}_{0}$ and subsequent air entrainment. However, at $z_{b}=0.022\unicode[STIX]{x1D706}_{0}$ , with less fluid momentum, a ripple instead of a micro-jet is generated and no air seems to be entrained. There possibly exists a boundary for the value of $z_{b}$ between the two conditions for the pressure oscillations to occur.

Figure 28.

Vertical velocity at the surface of the undisturbed breaker at the same streamwise position ( $x=0$ ) as the front surface of the structure versus time. The time $t_{s}$ is the instant when the wave maker starts moving. The vertical line on the left (dotted) represents the time ( $(t-t_{s})f_{0}=16.39$ ) when the breaker surface touches the lower front corner of the wall at condition $z_{b}/\unicode[STIX]{x1D706}_{0}=0.022$ . The vertical line on the right (dash-dot) represents the time ( $(t-t_{s})f_{0}=16.46$ ) when the breaker surface touches the lower front corner of the wall at condition $z_{b}/\unicode[STIX]{x1D706}_{0}=0.043$ . The bold dashed line is the polynomial fit to the velocity data. The values of $W_{0}$ at the above two times are found by interpolation of this polynomial.