### 1. Introduction

### 2. Theoretical Background

*p*denotes pressure,

*ρ*denotes fluid density,

*t*denotes time,

*x*and

*z*denote the Cartesian coordinate systems in the horizontal and vertical directions, respectively, and

*u*and

*w*denote the instantaneous velocities for

*x*and

*z*, respectively. The spatial and temporal acceleration terms of the fluid at time

*t*in Eqs. (1) and (2) were calculated by applying the central difference method (Eqs. (3), (4)) for the continuous velocity field measured through PIV, as shown in Fig. 1. where

*δt*denotes the time interval between images measured through PIV,

*h*denotes the space interval between each vector, and

*n*denotes the number of images between continuous velocity fields used in the acceleration calculation.

*x*-axis direction is integrated for an arbitrary point A on the free surface. Then, the pressure for an arbitrary point B on the L1 line is measured through integration in the

*x*direction and −

*x*direction along L2 perpendicular to L1. Based on the measured pressures on L1 and L2, the space marching integral (Eq. (7)) (Baur and Köngeter, 1999) is used to measure the pressure of the rest of the field, as shown in Fig. 3. Next, the pressure field for a location other than point A on the free surface was measured in the same way, and the average value of each location of the pressure field measured in each measurement process was used as the final pressure field. where

*S*denotes a reference position, and

_{ref}*s*denotes a spatial position to be calculated.

### 3. Wave-in-Deck Load Experiment Method

### 3.1 Experimental Conditions

*H*denotes the crest height,

_{c}*H*denotes the trough height,

_{t}*H*denotes the wave height,

*T*and

_{r}*T*denote the time taken for the wave to reach the crest from the average surface, and to reach the average surface from the crest, respectively, and

_{f}*T*and

_{zu}*T*denote the zero up-crossing and zero down-crossing periods, respectively. Each variable is schematically shown in Fig. 5.

_{zd}### 3.2 Pressure Measurement Method

### 3.3 PIV Measurement Setup and Method

*λ*) 532 nm] was used as a light source for the reflection of the particles.

*δt*= 500 Hz) for a field of 0.44 × 0.32 m

^{2}.

*C*denotes the cross-correlation function,

*M*and

*N*denote the number of pixels in the

*x*- and

*y*-direction interrogation area,

*i*and

*j*denote the target image coordinates, and

*f*and

*g*denote the particle distribution of the continuous image.

*U*

_{2}

*(*

_{D}*n*) denotes the surrounding vector,

*U*2

*D*(

*i*,

*j*) denotes the test vector, and

*∊*denotes the limit value used for the test, which was set to 1.1 in this study.

_{thresh}*d*/

_{τ}*d*by dividing the particle diameter (

_{pix}*d*) in the image by the pixel interval (

_{τ}*d*) (Prasad et al., 1992). The term

_{pix}*d*can be calculated as follows: where

_{τ}*M*denotes the ratio of the distance between the image and the lens, and the distance between the lens and the field of view;

*d*denotes the actual diameter of the particle; and

_{p}*d*denotes the diameter of the particle observed in the image by laser diffraction (Hecht and Zajac, 1974), which is derived from Eq. (11).

_{diff}*d*/

_{τ}*d*for the PIV measurement area used in this study was calculated to be approximately 0.09, and the corresponding measurement error was approximately 0.06 pixels (Raffel et al., 1998). In other words, it has a measurement error of approximately 2.55% of the local instantaneous maximum velocity (about 0.3 m/s) of the fluid measured through this PIV method.

_{pix}### 4. Experimental Results

### 4.1 Comparison of Pressure Estimation Results According to Time Interval

*t*= 2

*nδt*) between velocity fields used in the acceleration calculation. Therefore, the time interval selection for the pressure calculation should precede the pressure estimation for increased accuracy.

*δt*, 10

*δt*, and 20

*δt*) between the velocity fields for the acceleration calculation is shown in the time series in Fig. 8. The time (

*x*-axis) was nondimensionalized to period of the zero down-crossing (

*T*) of the free surface of the wave, and the pressure (

_{zu}*y*-axis) was nondimensionalized to

*ρgH*, as shown in Fig 8. The red dots are the pressure estimation results based on the PIV measurement velocity field, and the black solid line is the pressure measurement results measured by the pressure sensor in the model experiment. The pressure estimation results vary greatly depending on the change in

*δt*. When

*δt*is relatively small (Fig. 8(a), Δ

*t*= 2

*δt*), a result close to the peak value of the shock pressure can be measured that rises momentarily when the wave hits the structure, but the estimated pressure results after the peak pressure fluctuate significantly. Conversely, when

*δt*is relatively large (Fig. 8(c), Δ

*t*= 20

*δt*), the estimated pressure generally matches well with the pressure measured through the pressure sensor, but shows a significant difference in the maximum value of the pressure. In other words, it was found that the accuracy of the velocity field-based pressure estimation method measured through PIV decreased as the time interval between velocity fields for the acceleration calculation became longer or shorter. This is believed to be due to the phenomenon that the accuracy error increases when the time interval between the velocity fields is short, and the truncation error increases as the time interval increases when calculating the acceleration using the velocity field (van Oudheusden, 2013).

*p*denotes the pressure measured through the pressure sensor,

_{m}*p*denotes the estimated pressure using the PIV measurement results, and

_{c}*i*denotes the discretized time.

*n*) between the velocity fields set for the acceleration calculation at each of the five pressure sensor installation positions are compared, as shown in Fig. 9. The NRMSD had the lowest value for

*n*= 5 at all pressure sensor installation positions. This means that when the time interval between the velocity fields for acceleration calculation is 10

*δt*, the result of the estimated pressure based on the PIV velocity field best matches the pressure measured by the pressure sensor. Based on this result, it was found that the time interval between the velocity fields of the PIV-based pressure estimation method for the wave-in-deck load showed the smallest difference from the pressure measurement results for Δ

*t*= 10

*δt*. The NRMSD results proposed in this study are expected to show different results depending on the flow or pressure characteristics of each phenomenon to be estimated. In addition, it is determined that an appropriate time interval needs to be selected considering the characteristics of each phenomenon.

### 4.2 Pressure Field Estimation Results for the Case of Wave Impact Load

*δt*(

*n*= 5). Overall, the instantaneous velocity field based pressure estimation method applied in this study produced results that are in good agreement with the measurement results through the sensors for the local pressure caused by the wave-in-deck load. The estimated results were close to the peak values of the impact pressure that increased for the short moment of the wave-in-deck load at each pressure measurement location. Afterward, the negative pressure generated due to the gradually decreased pressure as the wave moved away showed good agreement overall. However, the pressure was relatively low for the pressure measured by the sensors for the peak impact pressure that increased rapidly; in particular, the difference was greater for P1 and P2 located at the leading edge of the deck. This seems to be attributable to the truncation error that occurred due to the relatively large time interval between the instantaneous velocity fields measured by PIV compared to the rise time of the pressure when the impact phenomenon occurred. In addition, it is believed to have some differences, since the impact load estimation method used in this study did not consider the viscosity of the fluid based on the Euler equation.

*x*-axis is the length of the structure (

*L*), the

*y*-axis is the depth of water (

*D*= 0.60 m), the velocity is

*L*/

*T*, and the pressure is

_{zu}*ρg H*nondimensionalized, as shown in Fig. 11.

### 5. Conclusion

*t*) between velocity fields for the acceleration calculation.When the time interval was short, the peak value of the instantaneously high impact pressure caused by the impact load was estimated well, but the overall pressure fluctuated due to an increase in the accuracy error. Moreover, the overall pressure was estimated well as the time interval increased, but the truncation error increased, resulting in a difference in the maximum value of the instantaneous impact pressure.