Physical Model Study

To validate the mathematical formulations presented in the earlier sections and to check key assumptions, use is made of small-scale hydraulic model tests reported previously by Cox and Scott (2001). Their experimental setup, shown schematically in Fig.1, and procedures are repeated here briefly. The hydraulic model experiment was conducted at Texas A&M University in a 36 m long by 0.95 m wide by 1.5 m high glass-walled wave flume equipped with a flap-type wavemaker capable of generating repeatable, irregular waves. The water depth in the tank was constant with $h$=0.80 m. The model deck consisted of a 0.935 m wide by 0.61 m long by 1.15 cm thick Plexiglas plate rigidly mounted to a steel frame and suspended from the top of the flume. The frame was constructed to minimize flow disturbance to the overtopping wave. Three cases with deck clearance heights of $z_c$=4.515, 6.015 and 8.515 cm were used in this study corresponding to the top deck of a fixed platform.

The free surface elevation, $\eta$, was recorded using a surface piercing wave gage (Gage 1) to quantify the free surface statistics in the vicinity of the deck. A second wave gage (Gage 2) was mounted on the deck at the leading edge and was used to measure the free surface elevation above the deck, $\eta_d$, during each overtopping event, where $\eta_d$=0 corresponds to the deck surface. A laser-Doppler velocimeter (LDV) was used to measure the instantaneous horizontal velocity, $u_{z_c}$, at an elevation $z$=2.0 mm above the deck. The overtopping rate per unit deck width was estimated as $q(t)$=$u_{z_c}\eta_d$, assuming that the velocity profile was uniform. Therefore, $u_{z_c}$ is regarded as $u_{max}$ in this paper. This $u_{max}$ is underestimated in comparison with real value obviously but the differences between them become small for high deck level case. The experimental procedure was shown by Cox and Scott (2001) to have excellent repeatability and the sensitivity of the measured overtopping to the LDV elevation above the deck was small.

Table 1: Summary of Free Surface and Overtopping Statistics for Cases 1 - 3.

Case	$z_c$	$h$	$N$	$H_{1/3}$	$T_{1/3}$	$N_o$	$\eta_{rms}$	$\mu_3$	$\mu_4$	$\alpha$	$Q_{rms}$
	(cm)	(cm)		(cm)	(s)		(cm)				(m$^2$)
1	4.515	80	3378	9.24	1.18	469	2.44	0.166	3.378	1.78	$5.414\times10^{-4}$
2	6.015	80	3323	9.54	1.19	252	2.52	0.144	3.346	1.78	$6.929\times10^{-4}$
3	8.515	80	3104	10.04	1.20	64	2.64	0.087	3.298	1.76	$6.770\times10^{-4}$

A JONSWAP wave spectrum with random phases was used to generate 20 statistically similar time series with a duration of 200 s each. This procedure of using several short time series rather than one long time series was adopted to minimize tank seiching and problems with wave reflection. All 20 runs were used for each of the three different deck clearances to assess sensitivity of the normalized exceedance probability distributions to the deck clearance. The combined time series contained over 3000 waves for each case, similar to the number of waves used in other studies of extreme wave statistics both in the laboratory (e.g. Kriebel and Dawson, 1993) and in the field (e.g. Forristall, 2000) as well as laboratory studies on the instantaneous overtopping of coastal structures (e.g. Franco and Franco, 1999). Table 1 summarizes the free surface and overtopping statistics where $z_c$ is the deck elevation, $h$ is the still water depth, $N$ is the number of incident waves measured at Gage 1, $H_{1/3}$ is the significant wave height at Gage 1, $T_{1/3}$ is the corresponding significant wave period, $N_o$ is the number of overtopping events recorded at Gage 2, $\eta_{rms}$ is rms value of the surface elevation at Gage 1, $\mu_3$ and $\mu_4$ are skewness and kurtosis of the surface elevation, $\alpha$ is the controlling parameter of the Weibull distribution calculated by maximum likelihood method, and $Q_{rms}$ is the rms value of overtopping volumes per unit deck width where each $Q$ is determined by integrating $q$ over the duration of each event. The relatively few overtopping events for Case 3 ($N_o$=64 as listed in Table 1) do not provide a robust statistical measure of the phenomena. Therefore, Case 3 is excluded from further analysis.

Following Cox and Scott (2001), a second laboratory model study with the same geometrical condition was conducted by Cox and Ortega (2002) in which they measured the detailed free surface and velocity for a single large wave overtopping the fixed deck. Fig. 2 shows the free surface profile of the incident wave measured at the same location in the flume corresponding to the leading edge of the deck both with and without the deck in place. There is only a small deviation in the total wave heights measured for the two cases (the wave crest amplitude with the deck in place is 3% larger than without the deck), and the effect of the structure on the total wave height is minimal. Thus, the deck is thin and has little influence on the incident wave.