Model/Data Comparisons and Discussions

Before comparing the model performance with the experimental data, it is necessary to check some of the basic assumptions used in the derivations. For example, the local wave number $k$ was regarded as a constant in the above formulation. Obviously, the assumptions of a locally sinusoidal wave crest profile and local wave number are valid for narrow banded spectra. Evidently from Eq.(12), the wavelength or period is important in estimating the overtopping for extreme waves. The relationship between wave period and wave height for random waves has a linear trend for small value of wave height, and the wave period becomes relatively constant for large amplitude waves. Therefore, the constant local wave number assumption is more crude for small amplitude waves but it is reasonably acceptable for large amplitude waves. Fig.8 shows experimental results of the relationship between $H$ and $T$. For large wave height, $T$ can be regarded as approximately constant, $T \simeq T_{1/3}$. Thus, the theory of overtopping is more accurate for larger amplitude waves and it is less valid for small amplitude waves. In this study, local wave number is assumed constant and $k$ is regarded as $k \simeq k_{1/3}$ calculated by the linear dispersion relationship with $T_{1/3}$. Of course, the assumption of $k \simeq k_{1/3}$ is not a necessary condition for the assumption of sinusoidal wave profile.

As mentioned above, the transformation of wave amplitude $A$ into overtopping volume $Q$ by Eq.(12) becomes worse as deck level $z_c$ increases. The model also assumes that the deck is thin and has little influence on the incident wave. Following dtc-scott2001, a second laboratory model study was conducted by dtc-ortega2002 in which they measured the detailed free surface and velocity for a single large wave overtopping the fixed deck. Fig. 9 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 height with the deck in place is 6% larger than without the deck), and the effect of the structure on the total wave height is minimal. To summarize the limitations of the theory, Eq.(15) and (16) are valid for 1) narrow banded wave spectra, 2) higher than middle deck levels, and 3) thin deck. Another unknown parameter, $\alpha$, is to be determined in Eq.(1) and (15). The estimation of $\alpha$ is beyond of this study, the value of $\alpha$ is directly calculated from the experimental data by maximum likelihood method in the following comparisons.

Figure 10: Model/data comparison of PDF of wave crest amplitude with experimental results (solid), Rayleigh wave amplitude distribution (dash) and Weibull wave amplitude distribution (chain) for (a) Case 1 and (b) Case 2.

=12cm

(a) Case 1.

=12cm

(b) Case 2.

Fig.10 compares the measured PDF of normalized wave crest amplitudes with the computed PDFs for the Rayleigh amplitude distribution and the Weibull amplitude distribution, Eq.(1). The values of $\alpha$ were directly calculated from the experimental data by the maximum likelihood method to be $\alpha$=1.78 for both Cases 1 and 2. The kurtosis values of the incident waves were about 3.4 and 3.3, respectively (Table 1). Therefore, the incident wave field was highly nonlinear compared to observations of real ocean waves. This difference may be to due to three dimensional or wind effects not present in the hydraulic model study. Since the values of $\alpha$ were smaller than 2.0, the peak of the Weibull distribution is lower than that of the Rayleigh distribution, and the tail of the Weibull distribution is higher. The Weibull distribution gives an overall better fit to the experimental data compared to the Rayleigh distribution.

Figure 11: Model/data comparison of exceedance probability of wave crest amplitude for (a) Case 1 and (b) Case 2 with Gage 1 ($\circ$), Gage 2 ($\times$), Rayleigh distribution at $z=0$ (dashed), Weibull distribution at $z=0$ (chain), Rayleigh distribution at $z=z_c$ (solid), Weibull distribution at $z=z_c$ (dotted).

=12cm

(a) Case 1.

=12cm

(b) Case 2.

Fig.11 compares the exceedance probability of the normalized wave crest amplitude for the experimental results at Gage 1 and 2 with the Rayleigh and Weibull distributions for Case 1 and 2. The exceedance probability of the Weibull distribution is calculated by Eq.(10) and the Rayleigh distribution is calculated by Eq.(10) with $\alpha=2$. The Weibull distribution gives better agreement to the experimental data for the seaward location (Gage 1) for both cases and for the waves on the deck (Gage 2) for the lower deck case (Case 1). For the higher deck case (Case 2), the Weibull distribution gives better agreement to the experimental data over all, however, the experimental data is located in between that of the Weibull and Rayleigh distributions in a few points at tail end of the data on the deck.

Figure 12: Model/data comparison of PDF of wave overtopping for (a) Case 1 and (b) Case 2 with experimental data (solid), modified Rayleigh overtopping distribution (dash), and modified Weibull overtopping distribution (chain).

=12cm

(a) Case 1.

=12cm

(b) Case 2.

Fig.12 compares the PDF of the normalized wave overtopping for the experimental results with the modified Rayleigh overtopping distribution Eq.(23), and the modified Weibull overtopping distribution Eq.(15). The modified Weibull overtopping distribution shows qualitatively better agreement with the experimental data than the modified Rayleigh overtopping distribution, particularly for larger values of $Q$. Fig.13 compares the exceedance probability of wave overtopping for the experimental results with the modified Rayleigh overtopping distribution Eq.(24) and the modified Weibull overtopping distribution Eq.(16). The modifiled Rayleigh overtopping distribution shows fair agreement with the experimental data for small values of $Q$, and the modified Weibull overtopping distribution shows particularly good agreement with the experimental data for larger values of $Q$. This is reasonable result because the main purpose to use the Weibull distribution is to estimate aqurately large wave events. However, the number of overtopping event is insufficient to compare quantitatively, but the qualitative trends of the overtopping volume can be estimated by Eq.(15) and (16).

Figure 13: Model/data comparison of exceedance probability of wave overtopping for (a) Case 1 and (b) Case 2 with experimental data (solid), modified Rayleigh overtopping distribution (dash), and modified Weibull overtopping distribution (chain).

=12cm

(a) Case 1.

=12cm

(b) Case 2.