3.1. Impact force and moment

In this subsection the normal component of the transient impact force and the transverse component of the moment during the impact of the three flexible plates under various values of $V_n$ and $UW^{-1}$ will be presented and discussed. The remaining force and moment components are at most 3 % and 10 % of the maximum values of $F_n$ and $M_{to}$, respectively, for the most extreme impact condition and are not presented here. The normal impact force, $F_n$, and the transverse moment about the plate centre, $M_{to}$, are plotted against time $t$ in figure 7(a,b), respectively, for the highest speed, $V_n=1.39\,{\rm m}\,{\rm s}^{-1}$ (${Fr} =0.43$, $R_D=1.33$), the thinnest plate ($h=6.61$ mm) and the four non-zero values of $UW^{-1}$. The time $t=0$ is chosen as the instant when the plate's trailing edge first makes contact with the still water surface. The large-scale features of the normal force and transverse moment curves are described as follows. The normal force increases at a variable rate over time during the impact until it reaches a maximum, ranging from approximately 2700 to 2900 N, and then suddenly decreases. The overall increase in force is qualitatively consistent with previous theoretical and experimental studies of vertical impact of rigid plates and wedges and is thought to be primarily due to the increasing wet plate area, which is under hydrodynamic pressure. The transverse moment, $M_{to}$, also initially increases with time but quickly reaches a maximum value before decreasing, crossing zero and reaching a sharp negative peak value ranging from $-400$ Nm to $-530$ Nm. The times of the negative peak values of $M_{to}$ match the times of the positive peak values of $F_n$ at the same impact conditions. As will be discussed in the following subsection, the times of these extreme values of $F_n$ and $M_{to}$ correspond to the times when the spray root reaches the leading edge of the plate, which is denoted in the following as $t=t_e$. At this instant, the leading edge of the plate begins to go under the local water surface. The temporal behaviour of $M_{to}$ is consistent with a temporally increasing magnitude of the total normal force and a centre of hydrodynamic pressure that moves from the trailing to the leading edge of the plate during the impact, as it does in the case of vertical impact of a rigid wedge or plate. Because of these competing effects, the moment reaches a maximum and then decreases to zero as the moment arm goes to zero when the centre of pressure moves across the middle of the plate. After this point, the moment becomes negative and continuously increases in magnitude (due to the increasing force and moment arm) until the spray root reaches the plate's leading edge; see § 3.2.

Figure 7. The normal force, $F_n$, and the transverse moment about the plate centre, $M_{to}$, are plotted vs time, $t$, in panels (a,b), respectively, for a single plate thickness ($h=6.61$ mm), $V_n=1.39\,{\rm m}\,{\rm s}^{-1}$ and various values of $U/W$. The time $t=0$ is the instant when the plate's trailing (low) edge first makes contact with the quiescent water surface. For these experimental conditions, ${Fr}=0.43$, $R_D= 1.33$ and $R_T$ ranges from 2.47 at $U/W = 8.33$ to 1.79 at $U/W = 4.5$.

In addition to the general features of the $F_n(t)$ and $M_{to}(t)$ curves in figure 7, as described above, several significant detailed features are noteworthy. Starting at early time, before $t=0$, the value of $F_n$ for all four cases undergoes small-amplitude oscillations with frequencies close to $f_{1a}$ $(= 25.1$ Hz), the first mode natural frequency of the plate in air; see table 2. These force oscillations are most likely due to plate oscillations excited by the prior horizontal and/or vertical accelerations of the carriage. The corresponding oscillations in the record of $M_{to}$ are less evident, indicating that the plate vibrations may be nearly symmetric, in support of the idea of first mode plate oscillations. After initial impact, as $F_n$ increases, there is a sudden decrease in slope at times increasing monotonically with increasing $U/W$, from $t \approx 0.05$ s at $U/W=4.5$ to 0.07 s at $U/W=8.33$. Close examination of the data indicates that the sharp change in slope appears to be part of a fine-scale oscillation in the curves, with a single period starting at approximately 0.03 s and lasting for approximately 0.08 s in all cases. The corresponding frequency ($\approx 20$ Hz) is a little less than $f_{1a} = 25.1$ Hz, which at this stage of approximately 20 % plate submergence seems an appropriate dynamic time scale. This slope change/oscillation cycle can also be observed as a slope decrease in the $M_{to}(t)$ curves during the same period of time. This oscillatory behaviour will be further examined in the discussion of the plate deflection in § 3.3. It also should be noted that, due to the small length of the wetted part of the plate, the flow is initially two dimensional and becomes more three dimensional as time proceeds. Thus, one would expect that the rate increase of force would change as time proceeds even for the case of the rigid plate. Subsequent to the instant of the break in slope in the $F_n$ plot, there are several cycles of small-amplitude secondary oscillations with frequencies approximately four times the plate's fundamental natural frequency in air. These oscillations damp out as $F_n$ rises over time. After these small high-frequency oscillations, the four curves of both $F_n$ and $M_{to}$ separate more significantly than before the break in slope occurs. In this region, the positive slope of the $F_n(t)$ curves increases with time as the peak value is approached. The times of the maximum $F_n$, the zero crossing of $M_{to}$ and the minimum of $M_{to}$ all increase monotonically with increasing $UW^{-1}$. Finally, the maximum value of $F_n$ is similar for all values of $UW^{-1}$, while the minimum value of $M_{to}$ increases as $UW^{-1}$ increases. Despite these differences, the shape and amplitudes of these curves are very similar, indicating that proper time scaling alone can collapse these curves significantly.

Nine plots of the inertially scaled dimensionless normal force, $F^*_n =F_n(\rho _wV_n^2BL)^{-1}$, and transverse moment, $M^*_{to}=M_{to}(\rho _wV_n^2BL^2)^{-1}$, vs dimensionless time $t/T_s$ are given in figures 8 and 9, respectively. In each of the three rows of plots in each figure, $V_n$ is held constant, resulting in one value of ${Fr}=V_n/\sqrt {gL}$ in each row, while in each column of three plots, $h$ is held constant, resulting in the same values of the natural period, $T_{1w}$, and bending stiffness, $D$. In each plot, curves for various values of $U/W$ are given. With this arrangement of plots based on the selection of $h$, $V_n$ and $U/W$ in the experiment, there results a single value of $R_D=\rho _wV_n^2L^3/D$ for each plot while each curve for a given value of $U/W$ within a plot corresponds to a different value of $R_T$. The values of these dimensionless ratios are given in the title of each plot. The raw data used to create figures 8(a i) and 9(a i) is the same as that plotted in figure 7(a,b), respectively.

Figure 8. The dimensionless normal force, $F^*_n =F_n(\rho _wV_n^2BL)^{-1}$, vs dimensionless time, $t/T_s$, is plotted for $V_n = 1.39$, 1.17 and 0.875, ${\rm m}\,{\rm s}^{-1}$ in rows (i), (ii) and (iii), respectively, and for three plate thicknesses $h = 6.61$ mm, 8.27 mm and 13.22 mm in columns (a,b,c), respectively. In each plot, the data for various values of $U/W$ are plotted, as indicated by the legends in (a i) for the plots in rows (i) and (ii) and in (a iii) for the plots in row (iii). The values of $h$, $V_n$, ${Fr}$, $R_D$ and $R_T$ are given above each plot; see table 4 for additional details. Results are shown for (ai)–(aiii) $h=6.61$ mm, (bi)–(biii) $h=8.27$ mm, (ci)–(ciii) $h=13.22$ mm.

Figure 9. The dimensionless transverse moment about the plate's centre, $M^*_{to}=M_{to}(\rho _wV_n^2BL^2)^{-1}$, vs $t/T_s$ is plotted for the same conditions as in the corresponding plots of $F_n^*(t/T_s)$ in figure 8. See the caption to figure 8 for additional details. Results are shown for (ai)–(aiii) $h=6.61$ mm, (bi)–(biii) $h=8.27$ mm, (ci)–(ciii) $h=13.22$ mm.

There are several notable features in the plots in figures 8 and 9. Perhaps the most obvious feature is that the four curves for $UW^{-1}\neq 0$ (each corresponding to a different value of $R_T$) are nearly collapsed to a single curve in each plot. This collapse includes fairly accurate alignment of the dimensionless time of the maximum $F^*_n$, the minimum $M^*_{to}$ and the zero crossing of $M^*_{to}$, and closely matched magnitudes of $F^*_n$ and $M^*_{to}$. The least successful collapse of the curves occurs over the dimensionless time range $0.2\leq t/T_s\leq 0.7$ in the cases with higher $V_n$. In this region of the $F_n^*(t/T_s)$ plots, the greatest vertical separation of the curves is approximately 7 % of the maximum $F^*_n$ for $V_n =1.39\,{\rm m}\,{\rm s}^{-1}$ and $h = 6.61$ mm in subplot (a i). Separations of similar magnitude are found in the $M^*_{to}(t/T_s)$ results. In general, the success of the alignment is due to the selection of $T_s$ to non-dimensionalize time and the selection of experimental conditions in which $V_n$ is held constant, rather than $U$ or $W$, which vary significantly from curve to curve in each plot. Also, the fact that $R_T$ varies continuously and monotonically from curve to curve over each row of plots (see the plot titles and table 4), while the collapsed curves change shape abruptly from plot to plot in rows (i) and (ii) (with a single value of $R_D$ for each plot), indicates that $R_D$, not $R_T$, is of primary importance in collapsing the data in each plot, at least for this range of $R_T$. When comparing the collapsed curves from plot to plot it can be seen that the overall magnitudes of the curves are fairly similar, a result that indicates that $\rho _w V_n^2$, which varies by a factor of 2.5 from row (iii) to row (i), is the primary scaling factor for forces and moments. However, the magnitudes and shapes of the $F^*_n$ and $M^*_{to}$ curves do have some trends from plot to plot. In particular, the peak values of the $F^*_n$ curves in the plots with the three highest values of $R_D$ (plots (a i), (a ii), (b i)) are slightly higher than in the other plots. Also, the shapes of the $F^*_n$ curves for these three highest values of $R_D$ have a region where the slope increases with time starting at approximately $t/T_s=0.2$ and ending with a sharp peak at $t_e/T_s$, while for the cases with lower $R_D$, the slope of the curves decreases with time throughout and the peak is less sharp. These variations in curve shape also include the change in slope in the $F_n(t/T_s)$ curves at approximately $t/T_s=0.2$ which is most evident in the curves for the two highest values of $R_D$ (plots (a i) and (a ii)), but gives way to a softer transition as $R_D$ decreases. As mentioned previously, the soft transition of the slope at small $R_D$ is likely a result of the transition from initially a nearly 2-D flow to a three-dimensional flow (see also Iafrati Reference Iafrati2016). It can also be seen that the peak values of $F^*_n$ and minimum values of $M^*_{to}$ occur at later $t/T_s$ as $R_D$ increases. All of these behaviours are clear evidence that at high $R_D$ the plate's flexibility significantly affects the magnitude and temporal evolution of the force and moment, a result that indicates a strongly coupled fluid–structure system. For the cases with the lowest values of $R_D$, the shapes of the force and moment curves are nearly independent of $R_D$. This indicates that at low $R_D$ the plate flexibility is less important in determining the forces and moments, i.e. that the deflections are small, even for the thinnest plate; see § 3.3 for verification of this conjecture. Finally, the curves for the $UW^{-1}=0$ in the plots in row (iii) at first seem anomalous; however, it should be noted that the jump in magnitude of $UW^{-1}$ from zero to the lowest non-zero value of $UW^{-1}$ is slightly greater than the range of non-zero values.

In order to better understand some aspects of the present $M_{to}(t^\prime )$ results, the times of the peak value, zero crossing and minimum value can be compared with the theoretical results of Wagner (Reference Wagner1932) (see also the discussion in the second paragraph of the introduction to this section) for the vertical water entry of a 2-D rigid wedge at infinite Froude number. The Wagner model predicts that the value of $M_{to}$ reaches a maximum at $t/T_s=0.25$ and crosses zero at $t/T_s=0.5$, with a minimum value occurring when the spray root reaches the plate's leading edge at $t/T_s = 2/{\rm \pi} = 0.637$. The present experimental conditions deviate from the Wagner theory since the plate motion is oblique (in most cases), the plate is flexible, the flow is somewhat three dimensional and the Froude number is not infinite, though in the first instants of the impact the wetted width-to-length ratio and Froude number based on wetted length are quite large. In the experiments the value of $t/T_s$ at the maximum $M_{to}$ is in the range $0.15\leq t/T_s \leq 0.2$, the $M_{to}$ zero crossings for oblique impacts occur in the range $0.5 \leq t/T_s \leq 0.7$, and the values of $t_e/T_s$ range from 0.75 to 0.85, with the later zero crossing and higher $t_e/T_s$ occurring for the highest $R_D$. In the vertical impact cases (see figure 9, row (iii)) the maximum $M_{to}$ occurs at $0.2 < t/T_s <0.3$, in better agreement with the Wagner's model than in the oblique impact cases; however, the zero crossing occurs even later in time than in the oblique impact cases.

The effects of $V_n$ and $h$ on $F^*_n(t/T_s)$ and $M^*_{to}(t/T_s)$ are illustrated more clearly by the plots in figure 10 where $U/W = 8.33$ for all curves. The figure contains plots of $F^*_n$ and $M^*_{to}$ vs $t/T_s$ for each of the three plate thicknesses. Each plot contains data for five values of $V_n$, resulting in separate values of $R_D$, $R_T$ and ${Fr}$ for each curve, as shown in the plot titles and legends. The scaling generally collapses the data in each plot to a thin band; however, in each of the plots for the two thicker plates, the curves for the four higher $V_n$ values collapse fairly accurately to a single curve; see plots (b i), (c i), (b ii) and (c ii). For the thinnest plate, the $F^*_n(t/T_s)$ and $M^*_{to}(t/T_s)$ data, plots (a i) and (a ii), respectively, form a single curve for small time, $t/T_s\lesssim 0.2$, but spread into separate curves, one for each combination of $R_D$, $R_T$ and ${Fr}$, as the peak force and minimum moment are approached at separate values $t_e/T_s$, for the four highest values of $V_n$. In these cases, the peak values of $F^*_n$ at first increase and then decrease, the minimum value of $M^*_{to}$ increases and the values of $t_e/T_s$ and $t/T_s$ at the $M^*_{to}$ zero crossing increase with increasing $R_D$ (decreasing $R_T$ and increasing ${Fr})$. The differences in the behaviour of $F^*_n(t/T_s)$ and $M^*_{to}(t/T_s)$ among the four highest values of $R_D$ for the thinnest plate are likely due to the influence of the dynamic response of the plate and the strong interaction between the plate deformation and the flow. For the lowest Froude number cases in figure 10, the curves move away from the collapsed data for $t/T_s\geq 0.2$. As mentioned previously, the effect of gravity becomes strong in these low Froude number cases, particularly at later times, and this is most likely the reason that the present inertial scaling fails. Further discussion of the low Froude number flow will be given in § 3.2.

Figure 10. The dimensionless normal force, $F_n^*$, and the dimensionless transverse moment about the plate centre, $M_{to}^*$, are plotted vs $t/T_s$ in rows (i) and (ii), respectively. The two plots in each column are for the same plate thickness (see below). All of the data in these plots is from runs with $U/W=8.33$ and each plot contains data for the same five values of $V_n$ as indicated by the ${Fr}$ values in the legend of plot (a i). The values of $R_D$ and $R_T$ in the legends and titles, respectively, of the top row of plots apply to the moment plots below in the same column. Results are shown for (ai) and (aii) $h=6.61$ mm, (bi) and (bii) $h=8.27$ mm, (ci) and (cii) $h=13.22$ mm.

In figure 11 plots of $F_n^*(t/T_s)$ (top row) and $M^*_{to}(t/T_s)$ (bottom row) are presented for plate motions with the same vertical velocity, $W = 0.57\,{\rm m}\,{\rm s}^{-1}$, and various values of $UW^{-1}$. The two plots in each column contain results for the same plate thickness. With this arrangement, each plot contains data for a single value of $R_T$ and curves with various values of $R_D$. For the two thicker plates, columns (b,c), the data for $U/W>0$ nearly collapse to single curves, while the $U/W=0$ curve is anomalous. These cases have relatively large values of $R_T$ (3.32 and 4.50 for plots (b) and (c), respectively) and relatively low values of $R_D$ (0.68–0.11 and 0.17–0.03 for plots (b,c), respectively). For the thinnest plate, column (a), the data does not collapse to a single curve indicating that the results are strongly influenced by flexibility for this value of $R_T (= 2.47$) and range of $R_D$ (1.33–0.22). The fact that the curves for conditions with the same relatively low value of $R_T$ vary significantly with $R_D$ confirms the above finding that $R_T$, while certainly important, is not dominant in determining the plate response.

Figure 11. The dimensionless normal force, $F_n^*$, and the dimensionless transverse moment about the plate centre, $M^*_{to}$, are plotted vs $t/T_s$ in the top and bottom rows of plots, respectively, for a single value of $W=0.57\,{\rm m}\,{\rm s}^{-1}$. The two plots in each column are for the same plate thickness (see plot titles). In each panel, the data for four Froude numbers are plotted: ${Fr}=0.43$, black line; ${Fr}=0.36$, blue line; ${Fr}=0.27$, red line; ${Fr}=0.17$, green line. These conditions are selected to yield a single value of $R_T$ in each column of two plots (see titles of the top row of plots) and the values of $R_D$ as listed in the legends of the top row of plots. Results are shown for (ai) and (aii) $h=6.61$ mm, (bi) and (bii) $h=8.27$ mm, (ci) and (cii) $h=13.22$ mm.

Some information about the shape of the pressure distribution on the plate can be learned from the plots of the dimensionless transverse moment arm about the plate's trailing edge, $M_{tt}(F_nL)^{-1}$, vs $t/T_s$ in figure 12. These plots are for the same conditions as those in figure 10. The dimensionless moment arm is the distance between the centre of pressure and the plate's trailing edge divided by $L$. In addition to the experimental data, two straight lines from theory are given in each plot. The dashed straight line is from the analytical solution by Wagner (Reference Wagner1932) for the infinite Froude number vertical water entry of a rigid 2-D wedge. This theory predicts that $M_{tt}(F_nL)^{-1}=t/T_s$. The dotted straight line is from the fictitious case of a uniform pressure distribution of magnitude $\rho V_n^2$ between the geometrical intersection of a rigid plate with the SWL and the plate's trailing edge. This case is a very low Froude number approximation and gives the relationship $M_{tt}(F_nL)^{-1}=0.5t/T_s$. For all plates and all values of $V_n$, the data for the evolution of the moment arm falls between the two extreme theoretical cases described above. For each experimental condition, after an initial transient, the moment arm curve follows the Wagner theory line and then falls off the Wagner line at values of $t/T_s$ that increase with increasing plate thickness and ${Fr}$. It is thought that for a rigid plate of the same dimensions as the plates in the present study, the moment arm curve would fall off the Wagner line at a $t/T_s$ slightly greater than that for the $h=13.22$ mm plate since, as will be discussed in § 3.3, the deflections for the 13.22 mm plate are small. It is also evident that the moment arm data for the thickest plate nearly collapses to a single curve when ${Fr}\geqslant 0.27$ while for the thinnest plate, the spread of the data is found to be more significant, as would be expected for strong fluid–structure interaction and considering the force and moment results in figure 10.

Figure 12. In each panel the dimensionless transverse moment arm about the plate's trailing edge, $M_{tt}(F_nL)^{-1}$, is plotted vs $t/T_s$ for $U/W=8.33$, a single plate thickness (as indicated in the title of each plot) and the same five values of $V_n$. The corresponding values of $R_D$, $R_T$ are indicated in the plot legends and titles, respectively. For the values of ${Fr}$, see the legend of plot (a i) in figure 10. Results are shown for (a) $h=6.61$ mm, (b) $h=8.27$ mm, (c) $h=13.22$ mm.

In order to summarize the effect of plate thickness on the impact force and moment, the dimensionless normal force, transverse moment about the plate centre and moment arm about the trailing edge are plotted against $t/T_s$ in figures 13(a), 13(b) and 13(c), respectively, for the most extreme experimental condition, ${Fr}=0.43$, $UW^{-1}=8.33$. Each plot contains three curves, one for each of the three plate thicknesses. As is shown in figure 13(a), the $F_n^*$ curves for all plate thicknesses nearly follow the same trend for $t/T_s\leqslant 0.2$. However, the one-cycle oscillation experienced by the thinnest plate at $t/T_s\approx 0.2$ and thought to be caused by the dynamic plate response at the initial impact stage is not obvious for the two thicker plates. Following the oscillation cycle described above, the value of $F_n^*$ for the thinnest plate falls off from that of the thicker plates. For a time period of $0.2\leqslant t/T_s \leqslant 0.7$, $F_n^*$ is smaller for the thinnest plate. Subsequently, before the time of maximum force, $F_n^*$ for the thinnest plate rises very quickly and surpasses values for the thicker plates. The separation of the data for $t/T_s>0.2$ also occurs in the dimensionless moment and moment arm curves, see figures 13(b) and 13(c), respectively. Both plots indicate that the more flexible the plate, the slower the propagation of the pressure centre. The dimensionless times for the pressure centre to reach the plate's centre for the thin, medium and thickest plates are approximately 0.72, 0.60 and 0.52, respectively, approaching the predicted value by the Wagner's 2-D rigid wedge model (0.5) for the thickest (most rigid) plate. Similarly, the moment arm propagation for the most flexible plate shows the most falloff from the 2-D rigid wedge model; see figure 13(c).

Figure 13. The dimensionless normal force, $F_n^*$ (panel a), the dimensionless transverse moment about the plate centre, $M_{to}^*$ (panel b), and the dimensionless transverse moment arm about the plate's trailing edge, $M_{tt}(F_nL)^{-1}$ (panel c), are plotted vs $t/T_s$ for $V_n = 1.39\,{\rm m}\,{\rm s}^{-1}$ (${Fr} =0.43$), $U/W$ $=8.33$ and each of the three plate thicknesses $h$ (as indicated by the legend in each plot). For these impact conditions, $T_s = 0.329$ s in all cases while $R_D = 1.33$, 0.68 and 0.17 and $R_T = 2.47$, 3.32 and 4.50 for $h= 6.61$, 8.27 and 13.22 mm, respectively, in each plot.

The final parameter of interest discussed in this subsection is the impulse of the normal force herein defined as

(3.6)\begin{equation} I_n=\displaystyle\int_{0}^{t_e} F_n\,{\rm d}t. \end{equation}

The values of $I_n$ from all experimental conditions with ${Fr}\geqslant 0.27$ are plotted vs ${Fr}$ in figure 14(a) and the corresponding values of the dimensionless impulse, $I_n^*= I_n/(\rho _wV_n^2BLT_s)$, are plotted vs $R_D$ ($=\rho _wV_n^2L^3D^{-1})$ in figure 14(b). For the remaining Froude number, ${Fr}=0.18$, $t_e$ is not well defined (see § 3.2) and this prevents the computation of an impulse comparable to those plotted in the figure. As can be seen in figure 14(a), $I_n$ increases with increasing ${Fr}$ and $UW^{-1}$, and to some degree with decreasing plate thickness. The range of the values of $I_n^*$ ($13.5 < I_n^* \leqslant 15$) is quite limited and the data form a nearly horizontal line when plotted against $R_D$ in figure 14(b). This limited range of $I^*_n$ can be traced back to the success of the $V^2_n$ scaling and changes in $t_e$ and the shapes of the $F_n(t/T_s)$ curves as the plate thickness is varied. As can be seen in figure 13(a), $t_e$ and the peak value of $F_n$ increase with decreasing plate thickness, thus tending to increase $I_n$. However, in the region $0.2 \leqslant t/T_s \leqslant 0.7$ the typical value of $F_n$ at a given time decreases with decreasing $h$. The changes in these competing effects largely balance each other in each case, resulting in similar values of $I_n^*$ for the three plate thicknesses.

Figure 14. (a) The impulse of the normal force, $I_n$, vs ${Fr}$ for conditions with $UW^{-1}>0$. (b) The dimensionless impulse, $I_n(\rho _wV_n^2BLT_s)^{-1}$, vs $R_D(=\rho _wV_n^2L^3D^{-1})$ for the same conditions as in plot (a). The impulse is the integral of the normal force, $F_n$, over the period from $t_I$ to $t_e$. Plotting symbols: $\circ$: $h=6.61$ mm; $\square$: $h=8.27$ mm; $\triangle$: $h=13.22$ mm. Plotting symbol colours: black, $UW^{-1}=8.33$; blue, $UW^{-1}=6.28$; red, $UW^{-1}=5.50$; green, $UW^{-1}=4.50$.

3.2. Water surface motion under the plate

In this section the under-plate water surface motion, which is characterized by the shape and motion of the under-plate spray root, is presented. In figure 15 four rows of three images from high-speed movies of the under-plate water surface evolution are shown. The three images in each set were taken at $t/T_s = 0.05$, 0.5 and 0.75. In the top three rows, the impact conditions are identical, ${Fr}=0.43$ and $UW^{-1}=8.33$, but the plate thickness varies with $h = 13.22$ mm, 8.27 mm, and 6.61 mm in rows (a), (b) and (c), respectively. This arrangement results in $R_D=$ 0.17, 0.68 and 1.33, and $R_T=$ 4.50, 3.32 and 2.47 in rows (a), (b) and (c), respectively. In row (d) images for ${Fr}=0.18$ and $UW^{-1}=8.33$ with $h=6.61$ mm ($R_D = 0.24$ and $R_T= 5.86$) are shown. The photographic set-up, interpretation of the images and extraction of the spray root line from these images is discussed in detail in § 2.5. The sequence of spray root lines, taken from the same high-speed movies as the images in rows (a) to (c) in figure 15, are plotted at uniformly distributed time steps in figure 16, where the profiles corresponding to the images in figure 15 are plotted in red. The profiles are plotted until the instant when any part of the spray root line emerges from the plate's leading edge.

Figure 15. Four sequences of three images from high-speed movies showing the spray root propagation under the plate for various conditions with $U/W=8.33$. The plate is moving from left to right and each column of images was taken at the same $t/T_s$ and with the same plate impact location relative to the camera, which remained at a fixed position under the tank for all measurements; see § 2.5 for details. The vertical symmetry wall, next to the port edge of the plate, is located near the lower side of each image, while the upper side of each image is facing the open towing tank. The values of $h$, ${Fr}$, $R_D$ and $R_T$ for each row of images is given in the titles of the images in the left column. The bright spots in the images in the left column are reflections of flood lights that are placed on the laboratory floor next to the high-speed movie camera. A composite movie showing the nine high-speed image sequences from which the images in the first three rows of this figure were taken is given as supplemental movie 1 available at https://doi.org/10.1017/jfm.2022.154. Results are shown for (ai)–(aiii) $h=13.22$ mm, ${Fr}=0.43$, (bi)–(biii) $h=8.27$ mm, ${Fr}=0.43$, (ci)–(ciii) $h=6.61$ mm, ${Fr}=0.43$, (di)–(diii) $h=6.61$ mm, ${Fr}=0.18$.

Figure 16. Three sequences of profiles of the spray root line measured along the lower surface of the plate as it bends during impact. The horizontal coordinate, $\xi$, is the longitudinal curvilinear distance from the plate's trailing edge, while the vertical coordinate, $\eta$, is the transverse curvilinear distance from the port edge, which is next to the vertical symmetry wall. The impact conditions and plate thicknesses are the same as those in the top three rows of images in figure 15. The time interval between successive profiles is 2.93 ms and the total time durations of the plotted profiles are 260.7 ms, 263.7 ms and 281.3 ms for (a,b,c), respectively. In each plot, the red profiles correspond to the photographs in figure 15, which are recorded at $t/T_s=0.05$, 0.50 and 0.75. The dashed line in each plot is the transverse location ($\eta = 0.125B$) where the spray root position vs time is recorded for further analysis. The spacing between successive profiles is proportional to the instantaneous spray root speed along the plate surface. Results are shown for (a) $h = 13.22$ mm, ${Fr}=0.43$, $UW^{-1}=8.33$, (b) $h = 8.27$ mm, ${Fr}=0.43$, $UW^{-1}=8.33$, (c) $h = 6.61$ mm, ${Fr}=0.43$, $UW^{-1}=8.33$.

Several qualitative features are apparent from the images and data in figures 15 and 16, respectively. At the early stage of the impact, the spray root lines for all four conditions are nearly straight and parallel to the plates’ trailing edges, indicating that the flow is nearly two dimensional at this early stage; see the images at $t/T_s=0.05$ in figure 15 column (i) and the corresponding profiles in figure 16. At later times, for example, $t/T_s=0.50$, the tangent of the spray root line is only parallel to the trailing edge near the plate's port edge, while the minimum longitudinal distance from the trailing edge to the spray root line appears at the plate's starboard edge. Thus, the three dimensionality of the flow increases as the instantaneous aspect ratio $B/\xi _r$, where $\xi _r$ is chosen as the curvilinear distance along the plate surface from the trailing edge to the spray root at $\eta = 0.125B$ (see caption to figure 16), of the wet area of the plate between the trailing edge and the spray root line decreases over time. Another interesting feature of the spray root behaviour is the effect of the plate thickness on the spray root's speed of propagation. This effect can be seen by comparing the profile sequences for the ${Fr}=0.43$ impacts for the three plates. At the early stage of the impact, the location of the spray root line relative to the plate's trailing edge is nearly the same for the three plates at corresponding times. At later times, for example, at $t/T_s=0.5$, the distance from the trailing edge to the spray root line decreases monotonically with increasing plate flexibility. Subsequently, the spray root emerges from the plate's leading edge at later times as the plate flexibility increases, as can be seen by comparing the images and profiles at $t/T_s = 0.75$. This difference in spray root speed can be seen in more detail in the spacing between successive spray root lines, which is proportional to the spray root speed. The spacing is approximately uniform over time for the thickest plate, while for the thinnest plate, the spacing first decreases to a minimum and then expands at the final stage of the impact, indicating that the propagation speed slows down and then speeds up. To a lesser degree, this behaviour can be seen in the profiles for the plate with intermediate thickness. The varying speed of the spray root is thought to be due to the plate's evolving deformation as will be discussed in § 3.3. One final feature of the data concerns the effect of Froude number as illustrated by comparison of the images for the thinnest plate between rows (c), where ${Fr}= 0.43$, and (d), where ${Fr}= 0.18$, of figure 15. As can be seen from the images at the same $t/T_s$ the spray root has travelled farther in the lower Froude number case. This may be associated with the smaller deflection in this case as will be shown in § 3.3. Also, in the last image in row (d), the spray root line has become irregular and partially obscure, in contrast to its sharp appearance in the other images. This behaviour is believed to be caused by the increasing influence of gravity as the impact proceeds, which is characterized by the decreasing instantaneous Froude number, $\mathit {Fr_t}=V_n(g\xi _r(t))^{-0.5}$, during the entire impact and ending with $\mathit {Fr_t} = {Fr}$. Under the strong influence of gravity toward the end of the impact in this case, observations indicate that the spray sheet over turns and falls back on the tank water surface immediately adjacent to the spray root. The impingement of the spray sheet on the tank water surface produces splashes and entrains clouds of air bubbles which can be seen in the image. Due to the effect of these phenomena on the flow approaching the plate from upstream, the spray root is not expected to follow the same physics as in the ideal potential flow analysis, such as Wagner's theory for a 2-D rigid wedge impact with infinite Froude number. In agreement with these ideas, the force and moment appear to deviate from the scaling that fits the high ${Fr}$ cases, as shown in figure 10 and discussed in § 3.1.

In the following, plots of $\xi _r/L$ at a fixed $\eta$ vs $t/T_s$ are used to compare the spray root propagation from one impact condition to another. The transverse position $\eta =0.125B$ was chosen for these measurements since this section of the spray root profile is typically farthest from and parallel to the trailing edge. This presentation is followed by a discussion of the connections between various features of the plots and the force and moment results from § 3.1.

Six plots of $\xi _r/L$ vs $t/T_s$ are shown in figure 17. As in the plots of $F^*_n(t/T_s)$ and $M^*_{to}(t/T_s)$ in figures 8 and 9, respectively, each plot contains curves for one plate thickness, single values of ${Fr}$ and $R_D$ and various values of $UW^{-1}$ and $R_T$. The conditions in rows (i) and (ii) of figure 17 are the same as those for the corresponding plots in rows (i) and (iii), respectively, of figures 8 and 9. One feature of the $\xi _r(t/T_s)/L$ curves that is common to these and all subsequent plots is that the curves follow the prediction of Wagner's theory for $t/T_s\lesssim 0.15$, (the dashed-dotted lines in the plots), but then fall below the theory line for later time. This initial agreement with the Wagner theory is to be expected since the assumptions of the theory (infinite Froude number, two dimensionality, rigid surface) are satisfied during the very early phase of impact; see above discussion. Another common feature of all of the plots in figure 17 is that the curves for the various values of $UW^{-1}$ nearly collapse to a single curve, as did the force and moment data in figures 8 and 9. The collapse to a single curve is more complete in the cases with ${Fr}=0.43$, row (i), $1.33 \geq R_D\geq 0.17$; however, this may be only a result of the larger range of $UW^{-1}$ in the plots for ${Fr}=0.27$, row (ii), $0.53 \geq R_D\geq 0.07$. Closer examination of the curves in each plot, particularly those for ${Fr}=0.27$, indicates that at the same $t/T_s$, the $\xi _rL^{-1}$ value is slightly greater, i.e. the spray root is closer to the plate's leading edge, for cases with greater $UW^{-1}$.

Figure 17. The dimensionless longitudinal position of the spray root at $\eta =0.125B$, $\xi _r/L$, is plotted vs $t/T_s$ for $V_n=1.39$ and 1.17 ${\rm m}\,{\rm s}^{-1}$ in rows (i) and (ii), respectively. The two plots in each column are for the same plate thickness (see plot titles). In each plot, data for various values of $U/W$ are plotted, as indicated by the legends. The experimental conditions for each plot are the same as those for the corresponding force and moment plots in rows (i) and (iii), respectively, of figures 8 and 9. The values of $h$, $V_n$, ${Fr}$, $R_D$ and $R_T$ are given above each plot. The straight dash-dotted (upper) lines represent the location of spray root as predicted by Wagner's theory for the infinite Froude number vertical impact of a rigid 2-D wedge. The straight dashed (lower) line represents the location of the geometrical intersection of the undeformed plate's lower surface with the still water surface. Results are shown for (a i) $h=6.61$ mm, (b i) $h=8.27$ mm, (c i) $h=13.22$ mm, (a ii) $h=6.61$ mm, (b ii) $h=8.27$ mm, (c ii) $h=13.22$ mm.

Comparison of the $\xi _r/L(t/T_s)$ curves from one Froude number/plate thickness condition to another (figure 17) yields a number of general trends. First, the curves in the three plots of row (ii), ${Fr}=0.27$ and $0.53 \geq R_D\geq 0.07$, are nearly the same. This is consistent with the previously described curves of $F_n(t/T_s)$ and $M_{to}(t/T_s)$ in row (iii) of figures 8 and 9, respectively. As will be shown in the following subsection, this is also consistent with the small plate deflection for all plate thicknesses at this low value of ${Fr}=0.27$. Second, in the three plots in row (i) (${Fr} =0.43$, $1.33 \geq R_D\geq 0.17$), the $\xi _r/L(t/T_s)$ curves are strongly affected by plate thickness, with $\xi _r/L$ at the same $t/T_s$ decreasing with decreasing $h$, increasing $R_D$ and decreasing $R_T$. Finally, comparisons of the pairs of plots for conditions with the same plate thickness and different ${Fr}$, $R_D$ and $R_T$ ranges, indicates that these parameters have little affect on the spray root propagation for the thickest plate ($R_D = 0.17$ and 0.07, $2.89\leq R_T \leq 7.14$), but significant affect for the thinnest plate ($R_D = 1.33$ and 0.53, $1.58\leq R_T \leq 3.91$), with the case with the highest $R_D$, plot (a i), exhibiting a significantly slower spray root propagation. Comparison of the shapes of the $\xi _r/L(t/T_s)$ and $F_n(t/T_s)$ curves in rows (i) of figures 17 and 8, respectively, indicates that the changes in the shape of the $F_n(t/T_s)$ curves as $h$ is decreased, including the increasing slope with time for $t/T_s>0.2$ and the very rapid increase in $F_n$ as $t$ approaches $t_e$, also appear in the $\xi _r/L(t/T_s)$ curves as the plate thickness is decreased.

In figure 18 three plots of $\xi _r(t/T_s)$ are shown with the experimental conditions for each plot the same as those for the corresponding plots of $F_n^*(t/T_s)$ and $M_{to}(t/T_s)$ in figure 10. Thus, the data in the three plots is for $UW^{-1}=8.33$ and each plot contains four curves for the same plate thickness and separate values of $V_n$. This arrangement results in the ranges of $R_D$ and $R_T$ varying from plot to plot as noted in the plot titles and legends. In all three plots, the values of $\xi _r/L$ at any $t/T_s\gtrapprox 0.15$ decrease with increasing $R_D$. As a result, at the same $t/T_s$, the spray root location is closer to the plate's leading edge for a case with smaller $R_D$ and the four curves in each plot separate. This trend is nearly negligible for the most rigid plate ($0.07 \leq R_D \leq 0.17$) and becomes more pronounced as the plate's relative flexibility increases ($0.53 \leq R_D \leq 1.33$ for the thinnest plate). The above described behaviour of the spray root trajectories is consistent with the various degrees of collapse of the $F^*_n(t/T_s)$ curves in the three corresponding plots in row (i) of figure 10.

Figure 18. Three plots of the dimensionless longitudinal position of the spray root, $\xi _r/L$, vs $t/T_s$ for $UW^{-1}=8.33$ and a single plate thickness for each plot, see plot titles. In each plot, the data for various values of $V_n$ are plotted. The experimental conditions are the same as in the corresponding force and moment plots in rows (i) and (iii), respectively, of figure 10. The values of $R_D$, $R_T$ and ${Fr}$ for each impact condition are given in the legends and titles of each plot. See the caption of figure 17 for the definitions of the two straight lines in each plot. Results are shown for (a) $h=6.61$ mm, (b) $h=8.27$ mm, (c) $h=13.22$ mm.

Three plots of $\xi _r(t/T_s)/L$ for the set of experiments with the same $W =0.57\,{\rm m}\,{\rm s}^{-1}$ are shown in figure 19. As in the corresponding $F_n^*(t/T_s)$ and $M_{to}^*(t/T_s)$ plots in figure 11, each plot in figure 19 is for a single plate and, therefore, since all conditions have the same $W$, a single value of $R_T$. Curves for the same set of $U/W$ values are shown in each plot, while the $R_D$ values vary from curve to curve and plot to plot. In the plots for $h=13.22$ mm ($R_T = 4.50$ and $0.03 \leq R_D \leq 0.17$) and $h=8.27$ mm ($R_T=3.32$ and $0.11 \leq R_D \leq 0.68$), the data nearly collapses to single curves which are also nearly identical from plot to plot. For the thinnest plate ($h=6.61$ mm, $R_T=2.47$ and $0.22 \leq R_D \leq 1.33$), the trajectories exhibit a slowing of the spray root motion with increasing $R_D$. This effect is relatively minor for $R_D = 0.22, 0.53$ and 0.94, but substantial for the curve for $R_D = 1.33$. This strong effect on the trajectory at the largest $R_D$ occurs in spite of the fact that all four curves were measured at conditions with the same value of $R_T$. Comparison of each of the three plots with the corresponding $F_n^*(t/T_s)$ and $M_{to}^*(t/T_s)$ plots in figure 11 reveals that while the force and moment curves for $U/W=0$ do not collapse with the curves for the other values of $U/W$, this anomalous behaviour is not present in the $\xi _r(t/T_s)$ data. As noted above, the $U/W=0$ condition corresponds to the lowest Froude number ($=0.17$) and from under-plate spray root movies, see the images in figure 15, it appears that the spray root structure is affected by gravity after the initial period of impact. This change in the flow may be responsible for the change in the force and moment curves; however, from figure  19(a) it appears that gravity has not affected the spray root propagation in a significant way.

Figure 19. Three plots of the dimensionless longitudinal position of the spray root, $\xi _r/L$, vs $t/T_s$ for a single value of $W=0.57\,{\rm m}\,{\rm s}^{-1}$ and a single plate thickness for each plot, see plot titles. In each plot, the data for four impact speeds are plotted and the corresponding Froude numbers are: ${Fr}=0.43$, black line; ${Fr}=0.36$, blue line; ${Fr}=0.27$, red line; ${Fr}=0.17$, green line. As in the corresponding plots of force and moment in figure 11, these conditions are selected to yield a single value of $R_T$ for each plate thickness. The values of $R_D$, $R_T$ and $U/W$ are listed in the legends and title of each plot. See the caption of figure 17 for the definitions of the two straight lines in each plot. Results are shown for (a) $h=6.61$ mm, (b) $h=8.27$ mm, (c) $h=13.22$ mm.

The effect of the plate thickness on $\xi _{r}(t/T_s)/L$ at three values of $V_n$ (0.88, 1.31 and 1.39 ${\rm m}\,{\rm s}^{-1}$) is illustrated by the plots in figures 20(a), 20(b) and 20(c), respectively. All of the data in the plots are for $U/W=8.33$ and within each plot there are three curves, one for each plate thickness. The data in plot (c) corresponds to the set of images shown in rows (a–c) of figure 15 and the spray root profiles in figure 16. At the lowest $V_n = 0.88\,{\rm m}\,{\rm s}^{-1}$, the spray root trajectories are nearly independent of plate thickness, probably as a result of the low relative flexibility at this low speed ($0.53 \geq R_D \geq 0.07$ and $3.91\leq R_T \leq 7.14$). However, as $V_n$ is increased, the curves separate and the shape of the trajectories for the two thinner plates at the two higher speeds ($1.33 \geq R_D \geq 0.61$ and $2.47\leq R_T \leq 3.51$) indicates that the spray root at first slows down and then speeds up in the final moments of the impact. These effects are illustrated in more detail in the plots in figures 21(a) and 21(b). In plot (a) the non-dimensional time delay relative to the Wagner theory value ($t_d/T_s = (t_{r}-t_W)/T_s$, where $t_{r}$ is the measured time for the spray root to reach position $\xi _r$ and $t_W = 2\xi _r \sin \alpha /({\rm \pi} W)$ is the time to reach position $\xi _r$ from Wagner's theory) is plotted as a function of dimensionless spray root position along the plate ($\xi _r/L$). The time delay starts out at zero but then begins to increase at $\xi _r/L \approx 0.2$. For the thinnest plate, the dimensionless time delay reaches 0.23 at $\xi _r/L \approx 0.95$ and then decreases slightly as the spray root reaches the plate's leading edge. In plot (b) the dimensionless spray root speed ($V_{r}/V_W$, where $V_{r}$ is the measured spray root speed (from the data in figure 20a) and $V_W = {\rm \pi}W/(2\sin \alpha )$ is the spray root speed in Wagner's theory) is plotted vs $\xi _r/L$. The dimensionless spray root speed is initially one but then, for the $h=6.61$ mm case, slows down to a minimum of approximately 0.55 at $\xi _r/L = 0.58$ and finally increases steadily to a maximum of 1.2 as the spray root reaches the plate's leading edge. This curve shape occurs, with decreasing amplitude, as $h$ increases.

Figure 20. Three plots of the dimensionless longitudinal position of the spray root, $\xi _r/L$, vs $t/T_s$ for $U/W=8.33$ and a single value of $V_n$ for each plot. Each plot contains three curves, one for each plate thickness: $h=6.61$ mm – black triangles ($\triangle$); $h=8.27$ mm – blue circles ($\circ$); and $h=13.22$ mm – red squares ($\square$). The conditions for plot (a) correspond to those in the plots of $F_n^*$, $M_{to}^*$ and $M_{tt}/(F_n L)$ in figure 13. See plot titles and legends for values of ${Fr}$, $R_D$ and $R_T$, and see the caption of figure 17 for the definitions of the two straight lines in the plots. Results are shown for (a) $V_n=0.88\,{\rm m}\,{\rm s}^{-1}$, ${Fr}=0.27$, (b) $V_n=1.31\,{\rm m}\,{\rm s}^{-1}$, ${Fr}=0.40$, (c) $V_n=1.39\,{\rm m}\,{\rm s}^{-1}$, ${Fr}=0.43$.

Figure 21. Characteristics of the spray root propagation for the most extreme impact condition, $V_n=1.39\,{\rm m}\,{\rm s}^{-1}$ and $UW^{-1}=8.33$, and the three plate thicknesses. (a) The dimensionless delay time relative to Wagner's infinite Froude number 2-D rigid wedge model, $t_dT_s^{-1}$, vs the longitudinal position of the spray root, $\xi _r L^{-1}$, where $t_d=t_{r}-t_W$, $t_{r}$ is the time for the spray root to reach a given position on the plate, and $t_W$ is the time to reach the same position according to Wagner's model. (b) The dimensionless speed of the spray root along the surface of the plate, defined as the ratio between the measured spray root speed $V_r = {\rm d}\xi _r/{\rm d}t$ and the spray root speed from Wagner's 2-D infinite Froude number wedge model ($0.5{\rm \pi} W/\sin \alpha$), vs $\xi _r L^{-1}$. The horizontal dashed and dash-dotted lines are the speed of the spray root from Wagner's model and the speed of the geometrical intersection between the lower surface of the undeformed plate and the SWL, respectively.

The dimensionless time delay when the spray root reaches the plate's leading edge ($t_d/T_s$ evaluated at $\xi _r/L = 1.0$, i.e. $t=t_e$) is plotted vs $R_D$ for all experimental conditions in figure 22. This time delay provides an overall measure of the effect of the plate thickness and impact conditions on the spray root propagation. As can be seen from the plot, the data generally follows a curve with a slope that increases slowly with increasing $R_D$. However, the data does form a vertical band of as much as $\pm 15\,\%$ about the mean at any $R_D$ and within this data, at each $R_D$, $t_d/T_s$ increases with increasing $U/W$ (increasing $R_T)$. Also, one can easily discern separate mean curves for each plate. These features indicate that $t_d/T_s$ is a complicated function of $R_D$, $R_T$, $U/W$ and ${Fr}$.

Figure 22. The dimensionless delay time at spray root emergence from the leading edge of the plate, $t_d/T_s$, vs $R_D$. The definition of $t_d$ is given in the caption of figure 21. Plotting symbol definitions: black – $h=6.61$ mm; blue – $h=8.27$ mm; red – $h=13.22$ mm; $\circ$ – $U/W=8.33$; $\square$ – $U/W=6.28$; $\diamond$ – $U/W=5.50$; $\triangle$ – $U/W=4.50$.

3.3. Plate out-of-plane deflection

In this section the plate out-of-plane deflection measurements from the five sensors distributed along the plate's centreline are presented and discussed in light of the force, moment and spray root results presented above. Before presenting these results we discuss the deflection measurement accuracy. Under all experimental conditions, the plate deflections are measured by monitoring the distance between the plate's upper surface and part of the frame connecting the dynamometer to the carriage; see § 2.4. Any compression of the structure between the plate and the fixed part of the deflection sensor will be added to the actual plate deflection to form the deflection reading. As the plate thickness increases, the plate deflections decrease and the carriage deflection, which depends primarily on the impact condition, will makeup a larger component of the deflection reading. Thus, in any experiment, for a large enough plate thickness, this error will overwhelm the actual plate deflection measurement. In the present experiments, we believe that at the most extreme impact condition ($V_n = 1.39\,{\rm m}\,{\rm s}^{-1}$, $U/W=8.33$) the carriage deflection might be as high as 1 mm. For this condition, the measured peak deflections at the centre of the plate are approximately 50 mm, 13 and 3 mm for the thin, medium and thick plates, respectively. In view of these results, we have not reported the data for the thickest plate since in this case the error could be as much as 50 % of the plate deflection.

In figure 23 sequences of instantaneous shape profiles of the two more flexible plates ($h=6.61$ mm and 8.27 mm) are plotted in a reference frame moving horizontally at speed $U$ (the horizontal carriage speed) for the impact conditions $V_n=1.39\,{\rm m}\,{\rm s}^{-1}$ and $UW^{-1}=8.33$. The geometrical intersection of the deformed plate's lower surface and the SWL is shown on each profile as is the instantaneous position of the spray root on the plate as obtained from the data in the previous subsection. Further details are given in the figure caption. By comparing the temporal evolution of the profiles of the two plates, it is evident that under the same impact condition, the thinnest plate experiences more significant deformation, as expected. The above-described moment arm and spray root results and the theory and experimental data for wedge vertical impact indicate that the force in the present experiments results from a high-pressure ridge followed by a plateau that moves longitudinally along the plate surface at non-constant speed. The plate deflection is, on the other hand, essentially a first mode response with maximum deflection at approximately the plate's midpoint for all time during the impact; see figure 23. In the case of the thinnest plate, there is a sudden pause in the plate motion which is seen as a decrease in spacing between the profiles centred at about the eighth profile ($t/T_s\approx 0.2$) from the top profile ($t/T_s = 0$). Thus, the pause occurs at the dimensionless time of the change in slope of the $F_n(t/T_s)$ curves; see figures 7(a) and 8(a i) for example. Since the bearing supports are moving vertically with constant speed, the intersection of the plate's lower surface and the SWL would move horizontally at constant speed if the plate were rigid. Thus, the plate deformation is also indicated by the non-uniform spatial distribution of the local geometrical intersection (blue squares, see figure caption) at the uniformly distributed times of the profiles in figure 23. For both plates, the spacing between intersection points first decreases (decreasing horizontal speed) and then increases (increasing horizontal speed). The spray root location, which is also noted in the figure (red circles), will be discussed at the end of this section.

Figure 23. Plate shape profiles are plotted in a reference frame that moves horizontally at speed $U$ for two cases with the same impact conditions, $V_n=1.39\,{\rm m}\,{\rm s}^{-1}$ and $UW^{-1}=8.33$, and two plate thicknesses: $h=6.61$ mm in plot (a) and $h=8.27$ mm in plot (b). The shape profile at each instant is estimated by a fourth-order polynomial fitted to the five plate deflection measurements and assuming that the deflection is zero at the position of the centrelines of the T-rails near the plate's leading and trailing edges. The profiles are measured with a frame rate of 1024 Hz and the time interval between successive profiles plotted here is 7.813 ms. The upper and lower dashed lines in each plot represent the undeformed plate positions when the trailing and leading edges reach the SWL ($z=0$), respectively. The solid and dash-dotted lines are profiles before and after the time ($t_e$) of spray root emergence, respectively. The instantaneous position of the spray root, determined by its optical projection on the plate as described in § 2.5 and shown previously in § 3.2, is marked by a red circle ($\circ$) on each profile. The local geometrical intersection of the SWL with the instantaneous plate's lower surface is marked by blue squares ($\square$). The horizontal and vertical axes are plotted with different scales, in order to illustrate the detailed features of the plate shape.

In the following, the deflection at the plate centre, $\delta _c$, is chosen to illustrate some typical behaviour as a function of impact parameters and plate thickness. The concepts of static and dynamic plate response as discussed relative to the simple beam model presented at the end of the introduction to § 3 will be used to help interpret the experimental results.

In each panel of figure 24, the dimensionless deflection at the plate centre, $\delta _c(R_DL)^{-1}=\delta _cD(\rho _w V_n^2 L^4)^{-1}$, is plotted vs $t/T_s$ for one value of $V_n$, one value of $h$ and various values of $UW^{-1}$. In each row of figure 24 the data for one of three values of $V_n$ (each row corresponding to a single value of ${Fr}$) are shown, while in each column, the data for one of the two thinner plates are shown. These plots correspond to the force, moment and spray root plots in figures 8, 9 and 17, respectively, though not all the conditions for each plot are the same. With this arrangement of experimental conditions, as in these earlier plots, each plot in figure 24 includes data for one value of $R_D$ and various values of $R_T$.

Figure 24. The dimensionless out-of-plane deflection at the plate centre, $\delta _c(LR_D)^{-1}$, is plotted vs $t/T_s$ for three typical values of $V_n = 1.39$, 1.31 and 1.17 ${\rm m}\,{\rm s}^{-1}$ in rows (i), (ii) and (iii), respectively, and for two plate thicknesses $h=6.61$ and 8.27 mm in columns (a,b), respectively. In each panel, the data for four values of $U/W$ are plotted, as indicated by the legend in plot (a i). The data in each plot is from single values of $R_D$ and ${Fr}$ and a range of values of $R_T$ as indicated in the plot titles. For $0\leq t/T_s \leq t_e/T_s$, the deflection is indicated by thick solid lines while, for $t/T_s>t_e/T_s$, thin dotted lines are used. These plots correspond to selected plots of $F_n^*(t/T_s)$, $M^*_{to}(t/T_s)$ and $\xi _r(t/T_s)$ in figures 8, 9 and 17, respectively. Results are shown for (a i) $V_n = 1.39\,{\rm m}\,{\rm s}^{-1}$, $h=6.61$ mm, (b i) $V_n = 1.39\,{\rm m}\,{\rm s}^{-1}$, $h=8.27$ mm, (a ii) $V_n = 1.31\,{\rm m}\,{\rm s}^{-1}$, $h=6.61$ mm, (b ii) $V_n = 1.31\,{\rm m}\,{\rm s}^{-1}$, $h=8.27$ mm, (a iii) $V_n = 1.17\,{\rm m}\,{\rm s}^{-1}$, $h=6.61$ mm, (b iii) $V_n = 1.17\,{\rm m}\,{\rm s}^{-1}$, $h=8.27$ mm.

A consistent feature of the plots in figure 24 for which $R_D \lesssim 1.0$ ((a iii) and all plots in column (b)) is that the four curves with various values of $U/W$ (and $R_T$) in each plot are nearly identical. Thus, as was found in all of the corresponding force, moment and spray root plots in figures 8, 9 and 17, respectively, the dimensionless ratios that depend on $V_n$ (${Fr}$ and $R_D$) dominate the plate response. For the two cases with $R_D\gtrsim 1.0$ (a i and a ii), the four curves separate in each plot with the maximum separation occurring in plot (a i), the cases with the highest $R_D$ and lowest values of $R_T$. The corresponding force, moment and spray root position curves, plots (a i,a ii) in figures 8, 9 and 17, respectively, have little variation from one $U/W$ to another.

Another important feature of the data is that both the dimensionless time and dimensionless magnitude of the peak deflection increase as $R_D$ increases from plot to plot going up in each column. This can be seen more clearly in figure 25 where curves of $\delta _c/(LR_D)$ vs $(t/T_s)$ are plotted for the cases with the four higher values of $R_D$ and $U/W=8.33$ for each plate thickness, with data for $h=6.61$ mm and 8.27 mm in plots (a,b), respectively. For the thicker plate, plot (b), $t/T_s$ at the peak deflection is relatively constant, increasing from approximately $0.5$ to $0.65$ as $R_D$ increases from 0.27 to 0.68 while the peak value of $\delta _c/(LR_D)$ increases from approximately $0.013$ to $0.017$ over the same range of $R_D$. For the thinnest plate, there is a wider variation of both quantities with $t/T_s$ at the peak deflection increasing from about $0.6$ to $0.9$ and $\delta _c/(LR_D)$ at the peak increasing from approximately $0.017$ to $0.037$ as $R_D$ increases from 0.53 to 1.33. The ranges of $R_T$ are given in the titles of each plot. It should be noted that for the thinnest plate and the highest $R_D$, the peak deflection occurs for $t > t_e$. For the cases in column (b), the fact that the time of maximum deflection approaches 0.5 for the smallest $R_D$ and that there are relatively small changes in the peak value of $\delta _c/(LR_D)$, indicate that the plate dynamics is tending to the static response case in this range of small $R_D$ and large $R_T$. For the cases in column (a), the rapidly increasing time and magnitude of the dimensionless maximum deflection, reaching $0.9$ and $0.037$, respectively, as $R_D$ approaches its highest values (and $R_T$ its lowest values), indicate a strong dynamic response. In these latter impact conditions, the unique shape of the curves of $F^*_n(t/T_s)$ at the highest values of $R_D$ (corresponding to the lowest values of $R_T$) as shown in figure 13 also indicate a strong two-way fluid–structure interaction. Given that the maximum deflection in the most extreme case ($\approx 50$ mm in plot (a) of figure 25) is approximately seven times $h$, the plate response is clearly nonlinear in these cases as well. It is thought that this nonlinearity may be partially responsible for the sensitivity of the deflection curves to $U/W$ in plots (a i,a ii) of figure 24.

Figure 25. The dimensionless out-of-plane deflection at the plate centre, $\delta _cL^{-1}$, vs $t/T_s$ for $U/W=8.33$ and $V_n = 0.88$ – 1.39 ${\rm m}\,{\rm s}^{-1}$ is plotted for two plate thicknesses: $h=6.61$ and 8.27 mm in panels (a) and (b), respectively. The values of $R_D$ and $R_T$ are indicated in the plot legends and titles, respectively. These plots correspond to selected plots of $F_n^*(t/T_s)$ and $M^*_{to}(t/T_s)$ in figure 10 and $\xi _r(t/T_s)$ in figure 18. For $0\leq t/T_s \leq t_e/T_s$, the deflection is indicated by thick solid lines while, for $t/T_s>t_e/T_s$, thin dotted lines are used. Results are shown for (a) $h=6.61$ mm, (b) $h=8.27$ mm.

One final feature of the plots in figure 24 is the pause in the deflection curves at $t/T_s=0.2$ in the cases with the thinnest plate. This pause corresponds to the pause in the motion of the plate deflection profiles as shown in figure 23(a). This behaviour includes a decrease of the slope of the centre point deflection curve. Also, the behaviour diminishes and disappears as $R_D$ decreases (due to increasing $h$ and/or decreasing $V_n$). This sudden pause in deflection occurs at the same time as the sudden jump in slope of the normal force curves at the same plate and impact conditions, as shown in figure 8 and discussed in § 3.1. For each case, the duration for the pause, considered as the duration of the hump between the two points with different nearly constant slopes just before and after the pause in both the $F_n$ and $\delta _c$ curves, approximately equals the natural period of the most flexible plate in air; see table 2. The appearance of this behaviour in the deflection curves lends further support to the hypothesis that the break in slope of the $F^*_n$ curves is associated with the dynamic response of the plate at the early stage of the impact, when most of the plate's lower surface remains in air and the effect of water added mass is thought to be relatively small.

To further explore the out-of-plane deflection results, the scaling of the maximum deflection at the plate centre point ($\delta _m$) is also discussed. To this end, $\delta _mL^{-1}$ is plotted vs ${Fr}^2$, $R_D$ and $R_T$ in figures 26(a), 26(b) and 26(c), respectively. Plotting the deflection vs ${Fr}^2$ is essentially using $V_n^2$ as the independent variable, since $g$ and $L$ are constants. In the plot of $\delta _mL^{-1}$ vs ${Fr}^2$ the data mainly forms two curves, one for each plate thickness. As pointed out in the discussion of figure 24, for the intermediate plate thickness, the maximum deflection is nearly independent of $UW^{-1}$ at all ${Fr}^2$, while for the most flexible plate, this conclusion is only true for the three lower values of ${Fr}^2$. At the two highest values of ${Fr}^2$, the most flexible plate undergoes greater deflection with larger $UW^{-1}$. In addition, the relationship between $\delta _mL^{-1}$ and ${Fr}^2$ is relatively linear for the intermediate plate, while for the most flexible plate, the relationship becomes highly nonlinear. It should be kept in mind that for ${Fr} \lesssim 0.18$, gravity affects the spray root as shown in § 3.2. As discussed previously, in a case with weak flow-structure interaction, the flow field is similar to that of the water entry of a rigid wedge, which is characterized by a self-similar moving pressure distribution proportional to the square of the normal impact speed, $V_n^2$. Under the assumption of a quasi-static deformation, the deformation should scale with the pressure distribution and be linear with $V_n^2$, i.e. ${Fr}^2$ since $g$ and $L$ are constants. The nonlinearity of the curve for the thinnest plate and mild nonlinearity for the moderate thickness plate in figure 26(a) indicates an increasing effect of the flow-structure interaction and dynamic plate response for the thinnest plate at the higher values of $V_n^2$, i.e. (${Fr}^2$). In the plot of $\delta _mL^{-1}$ vs $R_D$, panel (b), the data comes close to falling on a single curve. However, closer examination reveals again the spread in $\delta _mL^{-1}$ with variation in $U/W$ at high $R_D$ for the thinnest plate. Also, in the region where the data for the two plates overlap, $0\leq R_D \lesssim 0.7$, it can be seen that the data for the thinnest and the intermediate thickness plates fall on similar but slightly different curves. Thus, while $R_D$, which is equal to the ratio of the hydrodynamic force, $\rho _wV_n^2L^2$, to the bending stiffness, $D/L$, captures much of the physics in the plate impact process, other factors are important as well. From the plot of $\delta _mL^{-1}$ vs $R_T$ (c), it can be seen that though the data loosely conform to a curve shape of $\delta _m/L \propto (|R_T-2|)$, the scatter is quite extensive.

Figure 26. The maximum dimensionless out-of-plane deflection at the plate centre, $\delta _mL^{-1}$, for all impact conditions vs ${Fr}^2$ (subplot a), $R_D$ (subplot b) and $R_T$ (subplot c). Symbol definitions: black – $h=6.61$ mm; blue – $h=8.27$ mm. $\circ$ – $U/W=8.33$; $\square$ – $U/W=6.28$; $\diamond$ – $U/W=5.50$; $\triangle$ – $U/W=4.50$; $\ast$ – $U/W=0.00$.

Additional information at the instantaneous location of the spray root can be obtained from the evolution of the plate deflection profiles. In figure 27(a) the inclination of the plate surface at the location of the spray root is plotted vs $t/T_s$ for $V_n = 1.39\,{\rm m}\,{\rm s}^{-1}$ at four values of $UW^{-1}$ for the thinnest and intermediate thickness plates. The inclination at each time is obtained by differentiating the plate deflection polynomial functions, like those shown in figure 23, at the spray root location on each profile. In figure 27(b) the dimensionless normal component of the plate's impact velocity (in the laboratory reference frame) at the location of the spray root ($V_n+V_{nr})/V_n$) is plotted vs $t/T_s$ for the same impact conditions as in plot (a). In this dimensionless normal velocity, $V_{nr}$ is measured in the reference frame of the vertical carriage and is taken as the component of the plate's local velocity in the direction normal to the plate's undeformed surface. According to panel (a), for $t/T_s\lesssim 0.4$, which corresponds to a dimensionless spray root location $\xi _rL^{-1}\lesssim 0.5$, see figure 20(c), the local inclination at the spray root, $\alpha _r$, is greater than the pitch angle of the undeformed plate, $\alpha =10^\circ$, while for $t/T_s\gtrsim 0.4$ or $\xi _rL^{-1}\gtrsim 0.5$, $\alpha _r$ becomes smaller than $\alpha$. This transition point, i.e. where $\alpha _r = \alpha$, occurs near $\xi _rL^{-1}=0.5$ for all cases, indicating that at $t/T_s=0.4$ the instantaneous maximum deflection point for all cases appears very close to the plate middle point. As is shown in the plots for the more rigid plate, even at the highest $V_n$, the temporal variations of both $\alpha _r$ and ($V_n+V_{nr})/V_n$) are very small, within $2^\circ$ and 0.05, respectively. For the most flexible plate, however, the slope of the plate at the spray root varies in a much wider range, from 11$^\circ$ to $2^\circ$, and the value of ($V_n+V_{nr})/V_n$) reduces to as low as 0.82 for $0.4 \lesssim t/T_s \lesssim 0.6$. It is also found that ($V_n+V_{nr})/V_n$) vs $t/T_s$ is nearly independent of $UW^{-1}$ for both plates. For the less flexible plate, the collapse of data among different values of $UW^{-1}$ is also valid for $\alpha _r$ vs $t/T_s$; however, for the more flexible plate, the collapse of $\alpha _r$ vs $t/T_s$ only holds before approximately $t/T_s=0.5$. At later times, $\alpha _r$ is smaller for a greater $UW^{-1}$ at the same $t/T_s$. This latter effect is believed to be caused by the greater deformation at greater $UW^{-1}$, under the influence of the plate's dynamic response at these conditions, as shown in figure 24(a i).

Figure 27. Plate kinematic characteristics at the location of the spray root for the impact condition with $V_n=1.39\,{\rm m}\,{\rm s}^{-1}$ and $h=6.61$ mm (${Fr}=0.43$ and $R_D = 1.33$). In (a), $\alpha _r$, the local angle of inclination (relative to horizontal) of the plate at the instantaneous location of the spray root, is plotted vs $t/T_s$ while in (b), $V_n+V_{nr}$, the component of the plate velocity at the instantaneous location of the spray root and in the direction of the normal to the undeformed plate, is scaled by the normal impact speed $V_n$ and plotted vs $t/T_s$. Symbol definitions: black – $h=6.61$ mm; blue – $h=8.27$ mm; $\circ$ – $UW^{-1}=8.33$; $\square$ – $UW^{-1}=6.28$; $\diamond$ – $UW^{-1}=5.50$; $\triangle$ – $UW^{-1}=4.50$.

The above-described local plate behaviour near the spray root shown in figure 27 can be used qualitiatively with the steady 2-D theory of rigid wedge impact to elucidate some of the mechanisms in the fluid–structure interaction. In general, the deformation of the plate creates two opposite influences on the temporal evolution of the impact load. During roughly the first one half of the impact, the upward motion of the plate surface due to plate deflection delays the time when the spray root passes a given longitudinal location (see figure 20c), resulting in a reduction of the plate area that is under hydrodynamic pressure (the portion between the plate's trailing edge and the spray root). This slowing of the spray root motion may be due to the increased local angle of incidence, $\alpha _r$, during this first part of the impact as shown in figure 27(a) (Wagner's theory predicts that the spray root speed decreases with increasing deadrise angle). In addition, during this early phase of the plate deformation (up to the time of maximum deflection), the upward plate motion relative to the mounting points causes a reduced local plate impact speed ($V_n+V_{nr}$), as shown in figure 27(b). In Wagner's theory, this reduced speed combined with the larger local deadrise angle produces a lower hydrodynamic pressure on the plate surface. The two mechanisms dominate over the middle stage of the impact ($0.2< t/T_s<0.6$) and combine to produce a reduced $F_n$ as observed for the thinnest plate; see figure 13(a). On the other hand, after the spray root passes the instantaneous location of maximum deflection, the plate deformation tends to generate a reduced local plate inclination; see figure 27(a). When the spray root and its associated large pressure gradient travel to the region with $\alpha _r < \alpha$, a more severe local impact is created by a greater local peak pressure due to the greater dimensionless pressure $p(\rho _wV_n^2)^{-1}$ at smaller inclination angles, according to analysis of vertical rigid wedge impact (see, for example, Zhao & Faltinsen Reference Zhao and Faltinsen1993). In addition, the reduced inclination also results in a faster propagation speed of the spray root (see figure 21b), i.e. a greater expansion rate of the area under hydrodynamic pressure. Furthermore, during the late stage of the impact, $t/T_s>0.6$, for the thinnest plate, $(V_n+V_{nr})/V_n$ rises rapidly reaching approximately 1.0 at the end of the impact, as shown in figure 27(b). This indicates that the normal component of the local plate surface velocity returns to the rigid plate value, $V_n$, for the thinnest plate at these impact conditions. These two mechanisms tend to increase the impact load and are thought to be dominant in the final stage of the impact ($0.6< t/T_s< t_e$), as is indicated by the fast rising $F_n$ at this stage, as shown in figure 13(a). During the early stage of the impact ($t/T_s<0.2$), due to the under developed plate deformation, the dynamics described above are not significant.