Formulation of PDF maximum velocity at the crest

The overtopping volume is not only important for engineering purpose but also the dynamic properties, maximum velocity and overtopping rate, are significantly required for designing offshore structures. However, few studies have been investigated statistical property of the maximum velocity for random waves and there is no general theory to predict velocity profile above SWL. Thus the simple assumptions are required to statistically formulate this problem as a first step. The most simplest approach to predict the maximum velocity is assuming the linear and local sinusoidal wave assumptions to the wave dynamics near the crest. Using these assumptions it is possible to estimate the maximum horizontal wave velocity component as a function of the wave amplitude $A$.

For linear sinusoidal wave in deep-water, the surface elevation and horizontal velocity component have following relations.

$$\eta(x,t) = A\sin(kx-\omega t)$$ (20)

$$u(x,z,t) = A\omega{\mbox\rm e}^{kz}\sin(kx-\omega t)$$ (21)

Assuming wave steepness $kA$ is small enough and expanding ${\mbox\rm e}^{kz}$ up to $O((kA)^2)$ and $O(kA)$, the horizontal velocity component at the crest: can be approximated as

$$u_{max} = (1+kA)\omega A + O\left((kA)^2\right)$$ (22)

$$= \omega A + O\left(kA\right)$$ (23)

Truncated solution, Eq.(22) and Eq.(22), are 0.47% and 9.5% smaller than full solution, Eq.(21) at $z=A$, for the case of $kA=0.1$, respectively. The truncation of ${\mbox\rm e}^{kA}$ does not gives so significant error in comparison with the local sinusoidal assumption for random wave field. Normalize $u_{max}$ by $g$, $k$ and $\eta_{rms}$, Eq.(22) becomes

$$u_{max}' = \frac{u_{max}}{\eta_{rms}\sqrt{gk}}$$

$$= (1+k\eta_{rms}A')A' + O\left((kA)^2\right)$$ (24)

$$= A' + O\left(kA\right)$$ (25)

Figure 13: Model/data comparison of relationship between normalized wave crest amplitude $A'$ and maximum horizontal velocity with experimental results (solid circle), approximated solution 1 by Eq.(24) (solid line) and approximated solution 2 by Eq.(25) (dash line) for (a) Case 1 and (b) Case 2.

=11cm

(a) Case 1.

=11cm

(b) Case 2.

Before formulate the PDF of the maximum horizontal velocity model, it is necessary to check the assumptions above paragraph. A local sinusoidal velocity field with small wave steepness was assumed for random wave field for relatively large amplitude waves. To check these assumptions and truncation, Fig.13 compares the measured relationship between normalized wave crest amplitudes $A'$ and the maximum horizontal velocity with approximated solutions by Eq.(24) and Eq.(25). The approximated solution by Eq.(24) seems to follow close to the maximum edge of the measured data, particularly in Case 2. Another approximated solution by Eq.(25) is smaller than Eq.(24) and passes in the middle of the measured data. It means that the maximum horizontal velocity of measured data was smaller than predicted by the linear sinusoidal wave theory. Generally, the linear wave theory overestimates nonlinear random wave kinematics. Therefore these results are acceptable for us. On the other hand, the scatter of in the figure for lower deck case (Case 1) is larger than higher deck level case (Case 2). Specially, some small amplitude waves have very high speed horizontal velocity in Case 1. These are collapsing waves that break on the deck and shoot across the deck with very high speed. The rest of the waves can be regarded as truncated wave that is clippedj by the deck. Therefore, we have to take into the consideration both of them, but the collapsing waves are neglected in this paper.

Following this result, the assumptions of local sinusoidal wave theory for horizontal velocity component predicts the maximum value of horizontal velocity component on the deck, although, the assumptions cannot expect quantitatively agreement with the experimental data.

Solving Eq.(24) for $A$, we have

$$A' = \frac{1}{2\kappa}(\sqrt{4\kappa u_{max}'+1} - 1)$$ (26)

where $\kappa=k\eta_{rms}$ and is characteristic wave steepness. Thus, the maximum horizontal velocity at wave crest: $u_{max}$ is simply formulated as a function of wave crest amplitude $A$. Obviously, Eq.(26) is only valid linear sinusoidal wave in deep-water but we will apply this to random waves with large amplitude waves. Substituting Eq.(26) into Eq.(9) gives the PDF of maximum horizontal velocity.

$$p_{\ensuremath{u_{max}}\,}(\ensuremath{u_{max}^\prime}\,, z_c')d\ensuremath{u_{max}^\prime}\, = \frac{\alpha\beta}{\sqrt{4\kappa \ensuremath{u_{max}^\prime}\,+ 1... ... \sqrt{4\kappa \ensuremath{u_{max}^\prime}\,+ 1}- 1 ) \right]^{\alpha-1} \!\!\!$$

$$\hspace{-2.5cm} \times \exp\left\{ -\beta \left\{ \left[ \frac{1}... ...ight]^{\alpha} - z_c'^{\alpha} \right\} \right\} d\ensuremath{u_{max}^\prime}\,$$ (27)

Integrating Eq.(27) between $[0, \infty)$, we have the exceedance probability of on the deck given by

$$P_{\ensuremath{u_{max}}\,}(\ensuremath{u_{max}^\prime}\,,z_... ...\right) \right]^{\alpha} - z_c'^{\alpha} \right\} \right\}$$ (28)

Eq.(27) and (28) represent the maximum horizontal velocity: distribution based on a modified form of the Weibull wave amplitude distribution with local linear sinusoidal assumptions and is simply referred to as the ``modified Weibull $u_{max}$ distribution" hereafter for clarity. For the special case of $\alpha$=2 which is the Rayleigh condition for the wave amplitude distribution, Eq.(27) and (28) are simplified as

$$p^R_{\ensuremath{u_{max}}\,}(\ensuremath{u_{max}^\prime}\,, z_c')d\ensuremath{u_{max}}\, = \frac{1}{2\kappa\sqrt{4\kappa \ensuremath{u_{max}^\prime}\,+ 1}} \left( \sqrt{4\kappa \ensuremath{u_{max}^\prime}\,+ 1}-1 \right) \!\!\!$$

$$\hspace{-2.5cm}\times \exp\left\{ -\frac{1}{8\kappa^2} \left[ \le... ...1 \right)^2 - 4 \kappa^2 z_c'^2 \right] \right\} d\ensuremath{u_{max}^\prime}\,$$ (29)

$$P^R_{\ensuremath{u_{max}}\,}(\ensuremath{u_{max}^\prime}\,,... ...e}\,+ 1}- 1 \right)^2 - 4 \kappa^2 z_c'^2 \right] \right\}$$ (30)

Following the same manner, Eq.(29) and (30) represent the $u_{max}$ distribution at crest based on a modified form of the Rayleigh wave amplitude distribution and is simply referred to as the ``modified Rayleigh $u_{max}$ distribution" hereafter for clarity.

Figure 14: Dependence of the exceedance probability of maximum horizontal velocity on $\alpha$ with $z_c'$=1.0 and $\kappa$=0.1.

=11cm

Figure 15: Dependence of exceedance probability of maximum horizontal velocity on deck level $z_c'$ with $\alpha$=2.0 and $\kappa$=0.1.

=11cm

Figure 16: Model/data comparison of PDF of for (a) Case 1 and (b) Case 2 with experimental data (solid), modified Rayleigh distribution (dash), and modified Weibull distribution (chain).

=11cm

(a) Case 1.

=11cm

(b) Case 2.

Figure 17: Model/data comparison of exceedance probability of for (a) Case 1 and (b) Case 2 with experimental data (solid), modified Rayleigh distribution (dash), and modified Weibull distribution (chain).

=11cm

(a) Case 1.

=11cm

(b) Case 2.

Fig.14 shows the sensitivity of the exceedance probability of the modified Weibull distribution on $\alpha$ for the case of $z_c'$=1.0 and $\kappa$=0.1. With a change in $\alpha$ from $\alpha$=1.5 to $\alpha$=2, the exceedance probability of decreases one order of magnitude at $\ensuremath{u_{max}^\prime}\,\simeq 3.8$ and two orders of magnitude at $\ensuremath{u_{max}^\prime}\,\simeq 5.0$. The decrease is slightly faster than overtopping volume $Q$. This is the reason why the function in the exponential in Eq.(28) is proportional to $-\sqrt{\ensuremath{u_{max}}\,}/\kappa^\alpha$. The small characteristic wave steepness $\kappa$ decrease exceedance probability rapidly. Fig.15 shows the sensitivity of the exceedance probability of to the deck level $z_c'$ for the case of $\alpha$=2.0. With a change of $z_c'$ from $z_c'$=0 to $z_c'$=2, the exceedance probability decreases one order of magnitude at $\ensuremath{u_{max}^\prime}\,\simeq 2.6$ and two orders of magnitude at $\ensuremath{u_{max}^\prime}\,\simeq 5.3$.

Fig.16 compares the PDF of the normalized for the experimental results with the modified Rayleigh distribution, Eq.(29), and the modified Weibull distribution, Eq.(27). The both of theories for show qualitatively worse agreement with the experimental data. One of reason of the differences between the theory and the measured data is that the measured data include small value that cannot expect in the theory as shown in Fig.13. Fig.17 compares the exceedance probability of for the experimental results with the modified Rayleigh distribution, Eq.(30), and the modified Weibull distribution, Eq.(28). The exceedance probability of measured have two inflection points for Case 1 and the both of the modified Weibull and Rayleigh distribution cannot predict this tendency. The main reason of this strange distribution is some small amplitude waves have very high horizontal velocity in Case 1 as explained before. These are collapsing waves effects that break on the deck and shoot across the deck with very high speed. The number of collapsing waves on the deck is decreased as increasing deck height. Therefore, the modified Weibull distribution shows relatively good agreement with the experimental data for high deck Case 2 in comparison with Case 1, although there is significant difference in the low velocity region. Thus, the qualitative trends of high can be estimated by Eq.(27) and (28) for middle deck level case.