Experimental reconstruction of three-dimensional hydrodynamic loading in water entry problems through particle image velocimetry

Abstract

Predicting the hydrodynamic loading during water impact is of fundamental importance for the design of offshore and aerospace structures. Here, we experimentally characterize the 3D hydrodynamic loading on a rigid wedge vertically impacting a quiescent water surface. Planar particle image velocimetry is used to measure the velocity field on several planes, along the width and the length of the impacting wedge. Such data are ultimately utilized to estimate the 3D velocity field in the whole fluid domain, where the pressure field is reconstructed from the solution of the incompressible Navier–Stokes equations. Experimental results confirm that the velocity field is nearly 2D at the mid-span of the wedge, while the axial velocity along the length of the wedge becomes significant in the proximity of the edges. The variation of the fluid flow along the length of the wedge regulates the hydrodynamic loading experienced during the impact. Specifically, the hydrodynamic loading is maximized at the mid-span of the wedge and considerably decreases toward the edges. The method proposed in this study can find application in several areas of experimental fluid mechanics, where the analysis of unsteady 3D fluid–structure interactions is of interest.

Appendices

Appendix 1: PIV measurements’ accuracy

We study the sensitivity of PIV analysis to the size of the interrogation area and investigate the effect of temporal resolution on the estimated velocity field. With respect to the size of the interrogation area, we perform the PIV analysis in “PIVlab” by maintaining a 50 % overlap and employing three different window sequences where a finer, IA− ($32\times 32, 16\times 16, {\rm and}\,8\times 8$ pixels), and a coarser, IA$+$ ($128\times 128, 64\times 64, {\rm and}\,32\times 32$ pixels), windowing sequence are considered by halving or doubling the window size in the main text IA0 ($64\times 64, 32\times 32, {\rm and}\,16\times 16$ pixels), respectively (Foucaut et al. 2004). With respect to the temporal resolution, we downsample PIV data recorded at 4, 2, and 1 kHz, and we analyze 50 and 25 images for 2 and 1 kHz sampling, respectively. For brevity, we only report results at the mid-span of the wedge, $z=0\,{\rm mm}$, for the first repetition.

Figure 12 illustrates the PIV estimated vertical velocity in the proximity of the keel together with the wedge velocity measured using “Xcitex Proanalyst.” In particular, Fig. 12a, b demonstrate the effects of the size of the interrogation areas and temporal resolution on the velocity field in the proximity of the keel, respectively. Results reported in Fig. 12a show that the measured velocity fields are approximately independent of the size of the interrogation area. Moreover, findings in Fig. 12b suggest that aliasing errors due to the finite sampling frequency are not likely to be significant in these experiments (Eckstein and Vlachos 2009).

To better assess both these effects, we calculated the normalized root mean square error between data obtained by using the finer (IA−) or coarser (IA$+$) interrogation areas with respect to the reference one (IA0) and between the downsampled data (2 or 1 kHz) and the reference one (4 kHz). The maximum difference in both cases is less than 14 %. As also reported in Panciroli and Porfiri (2013) and Nila et al. (2013), we find that PIV measurements tend to underestimate the fluid velocity in the initial phase of the impact, which is likely due to presence of large velocity gradients close to the wedge, which are then smeared out through PIV (Panciroli and Porfiri 2013).

For completeness, in Fig. 13, we also present the hydrodynamic loading computed for three subsequent time instants: $t=9.75$, 10, and 10.25 ms on three different cross sections orthogonal to the z-direction and reconstructed from 2D analysis. The hydrodynamic loadings for neighboring time instants are very similar, suggesting that the selected PIV settings do not cause aliasing of the velocity measurements (Meinhart et al. 1993; Raffel et al. 2007).

Fig. 12

Comparison between the PIV measurement of the vertical velocity of the fluid in the proximity of the keel and the measured velocity of the wedge $\dot{\xi }$. a Effect of variation of the interrogation size. b Effect of the temporal resolution. Note that data in a are at 4 kHz, while data in b use the reference window sequence IA0

Fig. 13

Normalized hydrodynamic loading on the wetted wedge at three successive time instant. Black, red, and blue lines refer to $z=0$, 48, and 96 mm, respectively

Appendix 2: Repeatability of the experiments

To substantiate the repeatability of the experiments, we report the normalized root mean square error between the measured velocity fields of the second and third repetitions with respect to the first for the whole duration of the impact. Figure 14 presents the normalized root mean square error for cross-sectional velocity components u and v at three different locations; $z=0$, 48, and 96 mm and axial velocity component w at three different locations, $x=0$, 4, and 8 cm. The difference between the repetitions is always less than 8.32 % for the whole duration of impact.

Results reported therein demonstrate a good repeatability of the experiments, justifying the feasibility of the proposed approach to reconstruct the 3D pressure field in the whole fluid bulk from the estimated velocity field. Future work will seek to further reduce the uncertainty between repetitions by improving on the experimental design and refine the data analysis. Based on the results presented in the Fig. 14 and analysis conducted by Shao (2009), the effect of turbulence is unlikely to be significant.

Fig. 14

Normalized root mean square error between the second and third repetition with respect to the first for: a u, b v, and c w components of the velocity field, respectively. The solid and dashed lines refer to the second and third repetitions, respectively. Note that in c, we terminate the analysis at frame 45 for $x=8\,{\rm cm}$, due the limited resolution during the water splash

As a further analysis, we report the hydrodynamic force results for the three repetitions at each measurement location. Graphs in Fig. 15 compare the reconstructed hydrodynamic force exerted on the wedge at $z=0$, 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, and 96 mm. Results presented therein also confirm a good repeatability between trials. To quantify the error between the drop tests, we calculate the normalized root mean square error of the second and third repetition with respect to the first one. As shown in Table 1, the difference between trials is at most 18.70 %, which again corresponds to the edge of the wedge.

Table 1 Normalized root mean square error of the second and third repetition with respect to the first trial, for each measurement location along the wedge length

Fig. 15

Comparison between repetitions for the time trace of the force per unit length at all the 13 measurement locations in xy-plane: a $z=0\,{\rm mm}$, b $z=8\,{\rm mm}$, c $z=16\,{\rm mm}$, d $z=24\,{\rm mm}$, e $z=32\,{\rm mm}$, f $z=40\,{\rm mm}$, g $z=48\,{\rm mm}$, h $z=56\,{\rm mm}$, i $z=64\,{\rm mm}$, j $z=72\,{\rm mm}$, k $z=80\,{\rm mm}$, l $z=88\,{\rm mm}$, and m $z=96\,{\rm mm}$

Fig. 16

Axial velocity components estimated from the continuity equation along the wedge length at the keel ($x=0\,{\rm cm}$) for different time instants together with PIV data from the second sets of experiments at $x=0\,{\rm cm}$. Solid lines are PIV data, and squares refer to the results from the continuity equation

Appendix 3: Computation of the axial velocity component from the continuity equation

Here, we consider the possibility of using the continuity equation to estimate the axial velocity from the cross-sectional velocity components sampled at the 13 locations along the wedge length (Gelfgat 2014; Lang and Limberg 1999). For an incompressible flow, the continuity equation reads (Panton 1994; White and Corfield 1991)

$$\begin{aligned} \frac{\partial u(x,y,z,t)}{\partial x}+\frac{\partial v(x,y,z,t)}{\partial y}+\frac{\partial w(x,y,z,t)}{\partial z}=0 \end{aligned}$$

(4)

Based on Eq. (4), the derivative of the axial velocity with respect to z is estimated at each grid point on cross-sectional PIV images from the cross-sectional velocity components. Then, the axial velocity is calculated by integrating using a forward difference scheme along the z-direction, starting at the mid-span of the wedge, where we set the velocity to be in the xy-plane. Figure 16 reports the axial velocity components estimated using this approach for the first repetitions in the first set of experiments against PIV results. Results confirm that the axial velocity component increases in the vicinity of the edge. The axial velocity obtained from the continuity equation tends to overestimate the velocity computed using our approach, and such difference is likely due to the accumulation of error during the integration along the z-direction of the continuity equation.