- Email: [email protected]
Exceedance probability for wave overtopping on a fixed deck
Ocean Engineering 28 (2001) 707–721
Technical Note
Exceedance probability for wave overtopping on a fixed deck Daniel T. Cox *, Christopher P. Scott Coastal and Ocean Engineering Division, Department of Civil Engineering MS 3136, Texas A&M University, College Station, TX 77843-3136, USA Received 18 November 1999; accepted 19 January 2000
Abstract Detailed laboratory measurements were made of the instantaneous free surface elevation in front of a fixed deck and the instantaneous free surface elevation, velocity, and overtopping rate at the leading edge of the deck. The study showed that the exceedance probabilities for the normalized maximum instantaneous overtopping rate and the normalized overtopping volume were predicted by a simple exponential curve. The measured exceedance probability seaward of the deck compared well with the nonlinear theory of Kriebel and Dawson (Kriebel D.L., Dawson T.H., 1993. Nonlinearity in wave crest statistics. In: Proceedings Ocean Wave Measurement and Analysis. American Society of Civil Engineers, pp. 61–75). Conditional sampling of the crest heights seaward of the deck gave a normalized probability distribution similar to that of the maximum water level measured on the deck for each overtopping event. However, the values used to normalize each distribution were not the same. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Wave overtopping; Instantaneous overtopping; Crest heights; Exceedance probability; Rayleigh distribution; Nonlinear waves; Extreme statistics; Laboratory investigation; Offshore structures
1. Introduction Many offshore platforms have suffered from significant wave loading on their lower deck (Bea et al., 1999), and ships on single point moorings used for oil storage also suffer from “green water” on their decks during large storms. However, our
* Corresponding author. Tel.: +1-979-862-3627; fax: +1-979-862-8162. E-mail address: [email protected] (D.T. Cox). 0029-8018/01/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 9 - 8 0 1 8 ( 0 0 ) 0 0 0 2 2 - 6
708
D.T. Cox, C.P. Scott / Ocean Engineering 28 (2001) 707–721
understanding of the wave kinematics such as crest velocities, accelerations, and overtopping rates for extreme events remains rudimentary. Many recent advances have been made in predicting the extreme crest distributions in relation to the deck clearance or air gap problem. Determining the deck elevation above the calm water level is one of the most important aspects in the design of an offshore platform. Kriebel and Dawson (1993) presented a robust model which accounted for second-order wave steepness, and their model agreed well with laboratory and field data in both deep water and depth-limited conditions. Forristall (1997) presented a more complex second-order nonlinear model, including wave directionality, to predict extreme crest elevations and that model also showed good agreement with field measurements. Although the overtopping of shallow water coastal structures has been well studied, there have been very few studies on overtopping of a fixed deck in deep water with high wave steepness (i.e. storm conditions). This paper presents detailed laboratory measurements of the instantaneous free surface elevation in front of a fixed deck and the instantaneous free surface elevation, velocity, and overtopping rate at the leading edge of the deck. The two goals of this study were: (1) to determine experimentally the probability distributions of the extreme water level and wave overtopping; and (2) to determine to what extent existing theories developed for extreme crest statistics could be applied to the overtopping problem. Section 2 of this paper presents the experimental setup and procedure. Section 3 presents the analysis of the extreme wave statistics seaward of the deck and then an analysis of the overtopping statistics. Section 4 summarizes and concludes this paper.
2. Experiment 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 experimental setup is shown schematically in Fig. 1. Horsehair was used at the far end of the flume to minimize wave reflection. The model deck consisted of a 90 cm wide by 61 cm long by 1 cm thick Plexiglas plate rigidly mounted to a steel frame and suspended from the top of the flume. The steel frame was constructed to minimize flow disturbance to the overtopping wave. The height from the still water level to the bottom of the deck is defined as the deck clearance in this paper, denoted Hdc. Three cases with Hdc=3.5, 5.0 and 7.5 cm were used in this study which would correspond to the lower deck of an offshore platform. The water depth in the tank was constant with d=0.80 m. The free surface elevation, h, 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, hd, during each overtopping event, where hd=0 corresponds to the deck surface. A laser-Doppler velocimeter (LDV) was used to measure the instantaneous horizontal velocity, u, at an elevation z=2.0
D.T. Cox, C.P. Scott / Ocean Engineering 28 (2001) 707–721
709
Fig. 1. Experimental setup.
mm above the deck. The overtopping rate per unit deck width was estimated as q(t)=uhd, assuming that the velocity profile was uniform. The experimental repeatability and the sensitivity of the measured overtopping to the LDV elevation above the deck is discussed in Appendix A. 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. Each measured time series of h, hd, and u was truncated to eliminate transitional effects and were then combined. 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, 1997), as well as laboratory studies on the instantaneous overtopping of coastal structures (e.g. Franco and Franco, 1999).
3. Analysis 3.1. Extreme crest heights seaward of the deck The combined free surface time series at Gage 1 was analyzed using standard zero-upcrossing methodology (e.g. Goda, 1985), and the statistics are listed in Table
710
D.T. Cox, C.P. Scott / Ocean Engineering 28 (2001) 707–721
1. In this table, tdur is the duration of the combined truncated time series; N is the total number of waves; Hs is the significant wave height defined as the average of the one-third highest waves; Ts is the corresponding significant wave period; Hm0 is the spectral estimate of the significant wave height, Hm0=4.004√m0; Tp is the spectral peak period; k is the wavenumber calculated using 0.95Tp, the still water depth d, and the linear dispersion relation; and R is the parameter from Kriebel and Dawson (1993) representing the nonlinearity, R=kHm0. The mean period of the highest waves, 0.95Tp⯝1.18 s, was used rather than the peak period itself, consistent with the work of Kriebel and Dawson (1993). Table 1 shows that Hm0 and Hs differ by less than 0.7% which is not statistically significant. Although the 20 wavemaker time series were identical for each case, the statistics of characteristic wave heights Hm0 and Hs are not exactly the same for the three cases because of the wave–structure interaction which differed for the three deck elevations. The characteristic wave periods Tp and Ts were essentially unaffected by changes in the deck clearance. It is also noted that for each run, it was common to see one or two waves breaking in the flume far from the deck, indicating that the general conditions for the test were for high wave steepness. Fig. 2 shows the exceedance probability for the wave crests, hc, at Gage 1 normalized by Hm0. The measured exceedance probability is given as P⫽
i N+1
(1)
where i is the ith rank of the crest height and N is the number of waves listed in Table 1. The solid curve shows the underprediction of the extreme crest heights for a Rayleigh distribution given by
冋 冉 冊册
P⫽exp ⫺8
hc Hm0
2
(2)
as discussed by Forristall (1978). Kriebel and Dawson (1993) derived a correction for the nonlinear crest amplitudes which is given by
冋 冉 冊冉
P⫽exp ⫺8
hc Hs
2
冊册
1 hc 1⫺ R 2 Hs
2
(3)
where R is the parameter for nonlinearity. For this paper, Hm0 is used in place of Table 1 Summary of free surface statistics at Gage 1 Case
Hdc (cm)
d (cm)
tdur (s)
N
Hs (cm)
Ts (s)
Hm0 Tp (cm) (s)
k (1/cm)
R
1 2 3
3.5 5.0 7.5
80 80 80
3600 3600 3600
3378 3323 3104
9.24 9.54 10.04
1.18 1.19 1.20
9.20 9.49 9.97
0.02941 0.02941 0.02941
0.271 0.279 0.293
1.24 1.24 1.24
D.T. Cox, C.P. Scott / Ocean Engineering 28 (2001) 707–721
711
Fig. 2. Exceedance probability for normalized wave crests at Gage 1 for three cases with measurements (open circle), Rayleigh distribution (solid), and nonlinear model of Kriebel and Dawson (1993) (dash–dot).
712
D.T. Cox, C.P. Scott / Ocean Engineering 28 (2001) 707–721
Hs in Eq. (3) to be consistent in comparison to Eq. (2), although Kriebel and Dawson (1993) used Hs determined by zero-upcrossing. Eq. (3) is shown by the dash–dot curve in Fig. 2 and greatly improves the agreement. Kriebel and Dawson (1993) also presented a finite depth correction which had little effect for the present data because the relative water depth for the three cases was d/L⯝0.37 which is close to the deep water limit. Fig. 3 shows the exceedance probability for the wave crests at Gage 1 with crest heights normalized by the predicted crest height for that probability (e.g. Forristall, 1997). The open circles indicate crest heights normalized by the Rayleigh distribution, and the solid dots indicate the crest heights normalized by the nonlinear model of Kriebel and Dawson (1993). The extreme crest heights are underpredicted by the Rayleigh distribution in the range 1.3⬍hc/Rayleigh⬍1.4, whereas Forristall (1997) found the underprediction to be in the range 1.1⬍hc/Rayleigh⬍1.2 for multidirectional random waves measured in the field. This discrepancy may be due to wave directionality not included in the model study or wave–structure interaction which is likely to have been less for the field study. More importantly, the figure shows that the nonlinear model greatly improves the prediction of the extreme crest heights. For completeness, Fig. 4 shows the joint distribution of wave crest, hc, and trough, ht, for all waves in Case 1; and Fig. 5 shows the joint distribution of H and T. Figures for Cases 2 and 3 are similar and are not shown for brevity. The general trend of increasing wave crests and troughs is obvious in Fig. 4, and the figure shows that the largest wave crests do not coincide with the largest wave troughs and viceversa. The joint wave height–wave period distribution, shown in Fig. 5, is similar to that observed by many researchers (e.g. Goda, 1985). Although Table 1 shows that there was an effect of deck clearance on Hs and Hm0 at Gage 1 due to wave–
Fig. 3. Exceedance probability for wave crests at Gage 1 with crest height normalized by prediction of crest height for that probability. Open circles indicate normalization by linear Rayleigh prediction, dots indicate normalization by nonlinear model of Kriebel and Dawson (1993).
D.T. Cox, C.P. Scott / Ocean Engineering 28 (2001) 707–721
Fig. 4.
713
Joint distribution of wave crest and wave trough for all waves in Case 1.
Fig. 5. Joint distribution of wave height and wave period for all waves in Case 1. Large solid circle indicates (Hs,Ts).
structure interaction, Figs. 2, 4 and 5 and similar figures for Cases 2 and 3 indicate that this effect was small among the three cases and that the conditions seaward of the deck can be considered nearly the same for all three cases. 3.2. Maximum deck water level and overtopping The instantaneous overtopping rate per unit deck width was estimated as q(t)=uhd. To simplify the analysis, only the overtopping for which u⬎0 was considered. Measurements of u⬍0 generally occurred well after the passing of the wave crest
714
D.T. Cox, C.P. Scott / Ocean Engineering 28 (2001) 707–721
and |umin|¿umax. As a result, the “zero-crossing” was defined as the first data point for which q⬎0, and the duration of the overtopping event continued until q=0. The flow was carefully seeded for the LDV measurements, but there were still occasional dropouts (less than 0.5%) in the LDV signal which were obvious upon inspection of the record. These dropouts were replaced by a linear interpolation of the adjacent measured values. In other words, there could not be more than one overtopping event per wave period. Table 2 lists the overtopping statistics for the three cases. As expected, the number of overtopping events, No, decreases with increasing deck clearance, Hdc. The number of overtopping events for Cases 1 and 2 is 429 and 252, respectively, which is much less than the number of incident waves in Table 1. No, however, is not too dissimilar to the number of waves used by Kriebel and Dawson (1993) in their analysis of extreme crest heights in finite water depth. Furthermore, No is substantially larger than the minimum number of overtopping events (No=30) accepted by Franco and Franco (1999) in their study on irregular wave overtopping of a coastal structure. The number of overtopping events is smaller for Case 3 with No=64, and the reliability of the statistics for this case is lower. For each overtopping event the maximum free surface displacement above the deck, hdm, is recorded, and Column 4 lists the significant maximum free surface displacement, (hdm)s, where the subscript s indicates the average of the one-third highest events. Similarly, the maximum velocity for each overtopping event is used to estimate the significant maximum velocity, (um)s, and the maximum instantaneous overtopping rate for each overtopping event is used to estimate the significant maximum instantaneous overtopping rate, (qm)s. The instantaneous overtopping rate is integrated for each event to give the overtopping volume per unit deck width for that event, Q, which is used to estimate the significant overtopping volume per unit deck width, Qs. The values of (hdm)s, (um)s, (qm)s, and Qs are listed in Table 2 and used to normalize the respective exceedance probability distributions in the following figures. The total volume of overtopping water for an entire case, Qt, is found by summing the No overtopping volumes. Not surprisingly, Qt decreases as Hdc increases; however, Qs increases with increasing Hdc. This is because there is a much larger number of small overtopping events as the deck height decreases. It is possible to set a minimum threshold to determine a “significant” overtopping event as did Franco and Franco (1999), although this was effectively achieved in the present experiment by Table 2 Overtopping statistics Case
Hdc (cm)
No
(hdm)s (cm)
(um)s (cm/s)
(qm)s (m2/s)
Qs (m2)
Qt (m2)
1 2 3
3.5 5.0 7.5
469 252 64
4.0 3.8 3.1
42.9 59.3 71.2
0.0127 0.0193 0.0203
0.00106 0.00138 0.00140
0.228 0.154 0.041
D.T. Cox, C.P. Scott / Ocean Engineering 28 (2001) 707–721
715
setting the LDV measuring volume at a fixed distance above the deck. Overtopping events measured by the LDV were repeatable both in magnitude and phase; therefore, all measured overtopping events were deemed “significant”. It is noted that it would be desirable to determine relationships among Hdc, (qm)s, Qs, Hs and Ts but this is beyond the scope of this paper since essentially only one detailed time series of the instantaneous overtopping was measured for a given wave condition at each deck elevation. Fig. 6 shows the exceedance probability distribution for the maximum water level on the deck normalized by (hdm)s, listed in Table 2. The exceedance probability distribution for Gage 1 is also plotted on this figure using a conditional sampling technique where the crest elevations are defined as hc⫺Hdh and Hdh=Hdc+D is the deck height including the Plexiglas plate with D=1.0 cm thickness. Only values for which hc⫺Hdh⬎0 are considered. The figure shows that the distributions are fairly similar except for the largest crest elevations probably because of the interactions with the structure. Table 3 compares overtopping statistics with conditional crest statistics at Gage 1. In this table, N1 is the number of crests exceeding the threshold Hdh, and (hc⫺Hdh)s is the average of the one-third highest crests above the threshold used in the normalization in Fig. 6. The technique was also applied using Hdc, where N2 in Table 3 indicates the number of crests exceeding this threshold, and (hc⫺Hdc)s is the average of the one-third highest crests above this threshold. No and (hdm)s are repeated from Table 2 to facilitate comparison. Interestingly, N1⯝No for the three cases, but (hc⫺Hdh)s⬍(hdm)s presumably because of the effects of the plate thickness. Decreasing the threshold from Hdh to Hdc roughly doubled the number of events, but increased the significant values only slightly (approximately 6%). The curves in Fig. 6 show the equation P⫽exp[⫺2ca]
(4)
where a=2.0 (solid), 1.5 (dotted), and 1.0 (dash–dot) and c=hdm/(hdm)s for Fig. 6. Eq. (4) with a=2.0 would be expected for a Rayleigh wave height distribution (i.e. c=H/Hs). Interestingly, a=1.5 gives a reasonable prediction of the exceedance probability for the maximum water level on the deck, although a theoretical justification cannot be found at present. Fig. 7 shows the exceedance probability for the normalized maximum horizontal velocity. The predictions are made by Eq. (4) with c=um/(um)s and a=2.0 (solid), 1.5 (dotted), and 1.0 (dash–dot). For Case 1, the measured distribution does not follow any discernible trend, although Cases 2 and 3 may follow a Rayleigh distribution except for the most extreme velocities in Case 2. Fig. 8 shows the exceedance probability for the normalized maximum overtopping rate for each event normalized by (qm)s. The predictions are made by Eq. (4) with c=qm/(qm)s and a=1.0 (dash–dot). Despite the irregular nature of the velocity distribution shown in Fig. 7, the overtopping rate is predicted reasonably well using the simple exponential curve. Finally, Fig. 8 also shows that the exceedance probability for the normalized overtopping volume, Q/Qs, can be reasonably well predicted using the simple exponential curve as well.
716
D.T. Cox, C.P. Scott / Ocean Engineering 28 (2001) 707–721
Fig. 6. Exceedance probability for normalized maximum water level on deck with measurements on deck (open circle), measurements at Gage 1 with conditional sampling (solid dot), and predictions using Eq. (4) with a=2.0 (solid), 1.5 (dotted), and 1.0 (dash–dot).
D.T. Cox, C.P. Scott / Ocean Engineering 28 (2001) 707–721
717
Table 3 Comparison of overtopping statistics with conditional crest statistics at Gage 1 Case
Hdh (cm)
N1
(hc⫺Hdh)s (cm)
Hdc (cm)
N2
(hc⫺Hdc)s (cm)
No
(hdm)s (cm)
1 2 3
4.5 6.0 8.5
613 270 52
3.1 2.8 2.7
3.5 5.0 7.5
1070 466 111
3.3 3.0 2.8
469 252 64
4.0 3.8 3.1
4. Summary This study focused on detailed laboratory measurements of the instantaneous free surface elevation in front of a fixed deck and the instantaneous free surface elevation, velocity, and overtopping rate at the leading edge of the deck. Three deck clearances were used which would correspond to the lower decks of a platform. A large number of incident waves (3104⬍N⬍3378) and overtopping events (64⬍No⬍469) were used to increase the statistical reliability of the observations. The tests were conducted using a number of statistically similar shorter runs to minimize effects of tank seiching and re-reflection, and the deck was designed to minimize flow disturbances to the overtopping wave. This study showed that the exceedance probability distribution for the normalized wave overtopping rates and overtopping volumes can be predicted by a simple exponential curve, p(c)=exp(⫺2c). The exceedance probability for the normalized maximum water elevation on the deck could be predicted by the curve, p(c)=exp(⫺2c1.5) although there is no theoretical justification for this expression. This normalized probability distribution is not affected by changes in the deck clearance for the range of deck clearances in this study. The exceedance probability for the normalized maximum horizontal velocity on the deck did not follow any consistent trend for all three cases, although it appeared to follow a Rayleigh distribution of p(c)=exp(⫺2c2) for Cases 2 and 3, except for the extreme velocities measured in Case 2. Seaward of the deck, the nonlinear model of Kriebel and Dawson (1993) gave good agreement to the exceedance probability distribution for the wave crests. Conditional sampling of the crest heights seaward of the deck gave a normalized probability distribution similar to that of the maximum water level measured on the deck for each overtopping event. The values used to normalize each distribution were not the same, however, probably because of the interaction with the structure. Additional studies would be necessary to relate the significant overtopping statistics including (hdm)s, (um)s, (qm)s, and Qs to the incident wave conditions and deck height. Acknowledgements This work was sponsored by the National Science Foundation Career Award CTS9734109 and by the National Science Foundation Research Experience for Under-
718
D.T. Cox, C.P. Scott / Ocean Engineering 28 (2001) 707–721
Fig. 7. Exceedance probability for normalized maximum horizontal velocity on deck with measurements (open circle) and predictions using Eq. (4) with a=2.0 (solid), 1.5 (dotted), and 1.0 (dash–dot).
D.T. Cox, C.P. Scott / Ocean Engineering 28 (2001) 707–721
719
Fig. 8. Exceedance probability for normalized maximum instantaneous overtopping rate, qm/(qm)s (open circle); normalized volume of overtopping water, Q/Qs (solid dot); and predictions using Eq. (4) with a=1.0 (dash–dot).
720
D.T. Cox, C.P. Scott / Ocean Engineering 28 (2001) 707–721
graduates Program. The authors thank Jose Roesset of the Offshore Technology Research Center at Texas A&M University for the use of the laser-Doppler velocimeter, John Reed for technical support in conducting the experiments, and Dave Kriebel for valuable comments on this work.
Appendix A. Overtopping repeatability and sensitivity To assess the repeatability of the measured overtopping, Run 1 of Case 1 was repeated 10 times. Fig. 9 shows the ensemble average of the instantaneous overtopping rate, qa, in the top panel. The middle panel shows the detail in the range 109⬍t⬍116 s with qa indicated by a solid line and the envelope of one standard deviation indicated by the dash–dot line. Although the deviation of the overtopping (±50%) is much larger than the free surface elevation measured at Gage 1 (±2%) (figure not shown), it was acceptable for this study since estimates of the instantaneous overtopping rate can typically vary by a factor of 2. The bottom panel shows
Fig. 9. Measured overtopping for Case 1. Top panel: ensemble average of 10 runs. Middle panel: detail of top panel in the range 109⬍t⬍116 s with standard deviation envelope. Lower panel: instantaneous overtopping for Run 1 repeated with u measured at z=2.0 mm (solid), z=4.0 mm (dash–dot), z=8.0 mm (dashed).
D.T. Cox, C.P. Scott / Ocean Engineering 28 (2001) 707–721
721
the sensitivity of the instantaneous overtopping to the elevation of the LDV above the deck. Run 1 of Case 1 was repeated three times with LDV positioned at z=2, 4 and 8 mm above the deck. The figure shows that q generally increases with LDV elevation above the bed. However, given the scatter for a single run (shown in the middle panel) and given that the exceedance probability curves were generated using normalized values, the conclusions of this study are not sensitive to the LDV position.
References Bea, R.G., Xu, T., Stear, J., Ramos, R., 1999. Wave forces on decks of offshore platforms. J. Waterway, Port, Coastal, and Ocean Engineering 125 (3), 136–144. Forristall, G.Z., 1978. On the statistical distribution of wave heights in a storm. J. Geophysical Research 83 (C5), 2353–2358. Forristall, G.Z., 1997. Wave crest distributions: observations and second order theory. In: Proceedings Ocean Wave Kinematics, Dynamics and Loads on Structures. American Society of Civil Engineers, pp. 372–395. Franco, C., Franco, L., 1999. Overtopping formulas for caisson breakwaters with nonbreaking 3D waves. J. Waterway, Port, Coastal, and Ocean Engineering 125 (2), 98–108. Goda, Y., 1985. Random Seas and the Design of Maritime Structures. University of Tokyo Press, Tokyo. Kriebel, D.L., Dawson, T.H., 1993. Nonlinearity in wave crest statistics. In: Proceedings Ocean Wave Measurement and Analysis. American Society of Civil Engineers, pp. 61–75.
Technical Note
Exceedance probability for wave overtopping on a fixed deck Daniel T. Cox *, Christopher P. Scott Coastal and Ocean Engineering Division, Department of Civil Engineering MS 3136, Texas A&M University, College Station, TX 77843-3136, USA Received 18 November 1999; accepted 19 January 2000
Abstract Detailed laboratory measurements were made of the instantaneous free surface elevation in front of a fixed deck and the instantaneous free surface elevation, velocity, and overtopping rate at the leading edge of the deck. The study showed that the exceedance probabilities for the normalized maximum instantaneous overtopping rate and the normalized overtopping volume were predicted by a simple exponential curve. The measured exceedance probability seaward of the deck compared well with the nonlinear theory of Kriebel and Dawson (Kriebel D.L., Dawson T.H., 1993. Nonlinearity in wave crest statistics. In: Proceedings Ocean Wave Measurement and Analysis. American Society of Civil Engineers, pp. 61–75). Conditional sampling of the crest heights seaward of the deck gave a normalized probability distribution similar to that of the maximum water level measured on the deck for each overtopping event. However, the values used to normalize each distribution were not the same. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Wave overtopping; Instantaneous overtopping; Crest heights; Exceedance probability; Rayleigh distribution; Nonlinear waves; Extreme statistics; Laboratory investigation; Offshore structures
1. Introduction Many offshore platforms have suffered from significant wave loading on their lower deck (Bea et al., 1999), and ships on single point moorings used for oil storage also suffer from “green water” on their decks during large storms. However, our
* Corresponding author. Tel.: +1-979-862-3627; fax: +1-979-862-8162. E-mail address: [email protected] (D.T. Cox). 0029-8018/01/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 9 - 8 0 1 8 ( 0 0 ) 0 0 0 2 2 - 6
708
D.T. Cox, C.P. Scott / Ocean Engineering 28 (2001) 707–721
understanding of the wave kinematics such as crest velocities, accelerations, and overtopping rates for extreme events remains rudimentary. Many recent advances have been made in predicting the extreme crest distributions in relation to the deck clearance or air gap problem. Determining the deck elevation above the calm water level is one of the most important aspects in the design of an offshore platform. Kriebel and Dawson (1993) presented a robust model which accounted for second-order wave steepness, and their model agreed well with laboratory and field data in both deep water and depth-limited conditions. Forristall (1997) presented a more complex second-order nonlinear model, including wave directionality, to predict extreme crest elevations and that model also showed good agreement with field measurements. Although the overtopping of shallow water coastal structures has been well studied, there have been very few studies on overtopping of a fixed deck in deep water with high wave steepness (i.e. storm conditions). This paper presents detailed laboratory measurements of the instantaneous free surface elevation in front of a fixed deck and the instantaneous free surface elevation, velocity, and overtopping rate at the leading edge of the deck. The two goals of this study were: (1) to determine experimentally the probability distributions of the extreme water level and wave overtopping; and (2) to determine to what extent existing theories developed for extreme crest statistics could be applied to the overtopping problem. Section 2 of this paper presents the experimental setup and procedure. Section 3 presents the analysis of the extreme wave statistics seaward of the deck and then an analysis of the overtopping statistics. Section 4 summarizes and concludes this paper.
2. Experiment 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 experimental setup is shown schematically in Fig. 1. Horsehair was used at the far end of the flume to minimize wave reflection. The model deck consisted of a 90 cm wide by 61 cm long by 1 cm thick Plexiglas plate rigidly mounted to a steel frame and suspended from the top of the flume. The steel frame was constructed to minimize flow disturbance to the overtopping wave. The height from the still water level to the bottom of the deck is defined as the deck clearance in this paper, denoted Hdc. Three cases with Hdc=3.5, 5.0 and 7.5 cm were used in this study which would correspond to the lower deck of an offshore platform. The water depth in the tank was constant with d=0.80 m. The free surface elevation, h, 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, hd, during each overtopping event, where hd=0 corresponds to the deck surface. A laser-Doppler velocimeter (LDV) was used to measure the instantaneous horizontal velocity, u, at an elevation z=2.0
D.T. Cox, C.P. Scott / Ocean Engineering 28 (2001) 707–721
709
Fig. 1. Experimental setup.
mm above the deck. The overtopping rate per unit deck width was estimated as q(t)=uhd, assuming that the velocity profile was uniform. The experimental repeatability and the sensitivity of the measured overtopping to the LDV elevation above the deck is discussed in Appendix A. 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. Each measured time series of h, hd, and u was truncated to eliminate transitional effects and were then combined. 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, 1997), as well as laboratory studies on the instantaneous overtopping of coastal structures (e.g. Franco and Franco, 1999).
3. Analysis 3.1. Extreme crest heights seaward of the deck The combined free surface time series at Gage 1 was analyzed using standard zero-upcrossing methodology (e.g. Goda, 1985), and the statistics are listed in Table
710
D.T. Cox, C.P. Scott / Ocean Engineering 28 (2001) 707–721
1. In this table, tdur is the duration of the combined truncated time series; N is the total number of waves; Hs is the significant wave height defined as the average of the one-third highest waves; Ts is the corresponding significant wave period; Hm0 is the spectral estimate of the significant wave height, Hm0=4.004√m0; Tp is the spectral peak period; k is the wavenumber calculated using 0.95Tp, the still water depth d, and the linear dispersion relation; and R is the parameter from Kriebel and Dawson (1993) representing the nonlinearity, R=kHm0. The mean period of the highest waves, 0.95Tp⯝1.18 s, was used rather than the peak period itself, consistent with the work of Kriebel and Dawson (1993). Table 1 shows that Hm0 and Hs differ by less than 0.7% which is not statistically significant. Although the 20 wavemaker time series were identical for each case, the statistics of characteristic wave heights Hm0 and Hs are not exactly the same for the three cases because of the wave–structure interaction which differed for the three deck elevations. The characteristic wave periods Tp and Ts were essentially unaffected by changes in the deck clearance. It is also noted that for each run, it was common to see one or two waves breaking in the flume far from the deck, indicating that the general conditions for the test were for high wave steepness. Fig. 2 shows the exceedance probability for the wave crests, hc, at Gage 1 normalized by Hm0. The measured exceedance probability is given as P⫽
i N+1
(1)
where i is the ith rank of the crest height and N is the number of waves listed in Table 1. The solid curve shows the underprediction of the extreme crest heights for a Rayleigh distribution given by
冋 冉 冊册
P⫽exp ⫺8
hc Hm0
2
(2)
as discussed by Forristall (1978). Kriebel and Dawson (1993) derived a correction for the nonlinear crest amplitudes which is given by
冋 冉 冊冉
P⫽exp ⫺8
hc Hs
2
冊册
1 hc 1⫺ R 2 Hs
2
(3)
where R is the parameter for nonlinearity. For this paper, Hm0 is used in place of Table 1 Summary of free surface statistics at Gage 1 Case
Hdc (cm)
d (cm)
tdur (s)
N
Hs (cm)
Ts (s)
Hm0 Tp (cm) (s)
k (1/cm)
R
1 2 3
3.5 5.0 7.5
80 80 80
3600 3600 3600
3378 3323 3104
9.24 9.54 10.04
1.18 1.19 1.20
9.20 9.49 9.97
0.02941 0.02941 0.02941
0.271 0.279 0.293
1.24 1.24 1.24
D.T. Cox, C.P. Scott / Ocean Engineering 28 (2001) 707–721
711
Fig. 2. Exceedance probability for normalized wave crests at Gage 1 for three cases with measurements (open circle), Rayleigh distribution (solid), and nonlinear model of Kriebel and Dawson (1993) (dash–dot).
712
D.T. Cox, C.P. Scott / Ocean Engineering 28 (2001) 707–721
Hs in Eq. (3) to be consistent in comparison to Eq. (2), although Kriebel and Dawson (1993) used Hs determined by zero-upcrossing. Eq. (3) is shown by the dash–dot curve in Fig. 2 and greatly improves the agreement. Kriebel and Dawson (1993) also presented a finite depth correction which had little effect for the present data because the relative water depth for the three cases was d/L⯝0.37 which is close to the deep water limit. Fig. 3 shows the exceedance probability for the wave crests at Gage 1 with crest heights normalized by the predicted crest height for that probability (e.g. Forristall, 1997). The open circles indicate crest heights normalized by the Rayleigh distribution, and the solid dots indicate the crest heights normalized by the nonlinear model of Kriebel and Dawson (1993). The extreme crest heights are underpredicted by the Rayleigh distribution in the range 1.3⬍hc/Rayleigh⬍1.4, whereas Forristall (1997) found the underprediction to be in the range 1.1⬍hc/Rayleigh⬍1.2 for multidirectional random waves measured in the field. This discrepancy may be due to wave directionality not included in the model study or wave–structure interaction which is likely to have been less for the field study. More importantly, the figure shows that the nonlinear model greatly improves the prediction of the extreme crest heights. For completeness, Fig. 4 shows the joint distribution of wave crest, hc, and trough, ht, for all waves in Case 1; and Fig. 5 shows the joint distribution of H and T. Figures for Cases 2 and 3 are similar and are not shown for brevity. The general trend of increasing wave crests and troughs is obvious in Fig. 4, and the figure shows that the largest wave crests do not coincide with the largest wave troughs and viceversa. The joint wave height–wave period distribution, shown in Fig. 5, is similar to that observed by many researchers (e.g. Goda, 1985). Although Table 1 shows that there was an effect of deck clearance on Hs and Hm0 at Gage 1 due to wave–
Fig. 3. Exceedance probability for wave crests at Gage 1 with crest height normalized by prediction of crest height for that probability. Open circles indicate normalization by linear Rayleigh prediction, dots indicate normalization by nonlinear model of Kriebel and Dawson (1993).
D.T. Cox, C.P. Scott / Ocean Engineering 28 (2001) 707–721
Fig. 4.
713
Joint distribution of wave crest and wave trough for all waves in Case 1.
Fig. 5. Joint distribution of wave height and wave period for all waves in Case 1. Large solid circle indicates (Hs,Ts).
structure interaction, Figs. 2, 4 and 5 and similar figures for Cases 2 and 3 indicate that this effect was small among the three cases and that the conditions seaward of the deck can be considered nearly the same for all three cases. 3.2. Maximum deck water level and overtopping The instantaneous overtopping rate per unit deck width was estimated as q(t)=uhd. To simplify the analysis, only the overtopping for which u⬎0 was considered. Measurements of u⬍0 generally occurred well after the passing of the wave crest
714
D.T. Cox, C.P. Scott / Ocean Engineering 28 (2001) 707–721
and |umin|¿umax. As a result, the “zero-crossing” was defined as the first data point for which q⬎0, and the duration of the overtopping event continued until q=0. The flow was carefully seeded for the LDV measurements, but there were still occasional dropouts (less than 0.5%) in the LDV signal which were obvious upon inspection of the record. These dropouts were replaced by a linear interpolation of the adjacent measured values. In other words, there could not be more than one overtopping event per wave period. Table 2 lists the overtopping statistics for the three cases. As expected, the number of overtopping events, No, decreases with increasing deck clearance, Hdc. The number of overtopping events for Cases 1 and 2 is 429 and 252, respectively, which is much less than the number of incident waves in Table 1. No, however, is not too dissimilar to the number of waves used by Kriebel and Dawson (1993) in their analysis of extreme crest heights in finite water depth. Furthermore, No is substantially larger than the minimum number of overtopping events (No=30) accepted by Franco and Franco (1999) in their study on irregular wave overtopping of a coastal structure. The number of overtopping events is smaller for Case 3 with No=64, and the reliability of the statistics for this case is lower. For each overtopping event the maximum free surface displacement above the deck, hdm, is recorded, and Column 4 lists the significant maximum free surface displacement, (hdm)s, where the subscript s indicates the average of the one-third highest events. Similarly, the maximum velocity for each overtopping event is used to estimate the significant maximum velocity, (um)s, and the maximum instantaneous overtopping rate for each overtopping event is used to estimate the significant maximum instantaneous overtopping rate, (qm)s. The instantaneous overtopping rate is integrated for each event to give the overtopping volume per unit deck width for that event, Q, which is used to estimate the significant overtopping volume per unit deck width, Qs. The values of (hdm)s, (um)s, (qm)s, and Qs are listed in Table 2 and used to normalize the respective exceedance probability distributions in the following figures. The total volume of overtopping water for an entire case, Qt, is found by summing the No overtopping volumes. Not surprisingly, Qt decreases as Hdc increases; however, Qs increases with increasing Hdc. This is because there is a much larger number of small overtopping events as the deck height decreases. It is possible to set a minimum threshold to determine a “significant” overtopping event as did Franco and Franco (1999), although this was effectively achieved in the present experiment by Table 2 Overtopping statistics Case
Hdc (cm)
No
(hdm)s (cm)
(um)s (cm/s)
(qm)s (m2/s)
Qs (m2)
Qt (m2)
1 2 3
3.5 5.0 7.5
469 252 64
4.0 3.8 3.1
42.9 59.3 71.2
0.0127 0.0193 0.0203
0.00106 0.00138 0.00140
0.228 0.154 0.041
D.T. Cox, C.P. Scott / Ocean Engineering 28 (2001) 707–721
715
setting the LDV measuring volume at a fixed distance above the deck. Overtopping events measured by the LDV were repeatable both in magnitude and phase; therefore, all measured overtopping events were deemed “significant”. It is noted that it would be desirable to determine relationships among Hdc, (qm)s, Qs, Hs and Ts but this is beyond the scope of this paper since essentially only one detailed time series of the instantaneous overtopping was measured for a given wave condition at each deck elevation. Fig. 6 shows the exceedance probability distribution for the maximum water level on the deck normalized by (hdm)s, listed in Table 2. The exceedance probability distribution for Gage 1 is also plotted on this figure using a conditional sampling technique where the crest elevations are defined as hc⫺Hdh and Hdh=Hdc+D is the deck height including the Plexiglas plate with D=1.0 cm thickness. Only values for which hc⫺Hdh⬎0 are considered. The figure shows that the distributions are fairly similar except for the largest crest elevations probably because of the interactions with the structure. Table 3 compares overtopping statistics with conditional crest statistics at Gage 1. In this table, N1 is the number of crests exceeding the threshold Hdh, and (hc⫺Hdh)s is the average of the one-third highest crests above the threshold used in the normalization in Fig. 6. The technique was also applied using Hdc, where N2 in Table 3 indicates the number of crests exceeding this threshold, and (hc⫺Hdc)s is the average of the one-third highest crests above this threshold. No and (hdm)s are repeated from Table 2 to facilitate comparison. Interestingly, N1⯝No for the three cases, but (hc⫺Hdh)s⬍(hdm)s presumably because of the effects of the plate thickness. Decreasing the threshold from Hdh to Hdc roughly doubled the number of events, but increased the significant values only slightly (approximately 6%). The curves in Fig. 6 show the equation P⫽exp[⫺2ca]
(4)
where a=2.0 (solid), 1.5 (dotted), and 1.0 (dash–dot) and c=hdm/(hdm)s for Fig. 6. Eq. (4) with a=2.0 would be expected for a Rayleigh wave height distribution (i.e. c=H/Hs). Interestingly, a=1.5 gives a reasonable prediction of the exceedance probability for the maximum water level on the deck, although a theoretical justification cannot be found at present. Fig. 7 shows the exceedance probability for the normalized maximum horizontal velocity. The predictions are made by Eq. (4) with c=um/(um)s and a=2.0 (solid), 1.5 (dotted), and 1.0 (dash–dot). For Case 1, the measured distribution does not follow any discernible trend, although Cases 2 and 3 may follow a Rayleigh distribution except for the most extreme velocities in Case 2. Fig. 8 shows the exceedance probability for the normalized maximum overtopping rate for each event normalized by (qm)s. The predictions are made by Eq. (4) with c=qm/(qm)s and a=1.0 (dash–dot). Despite the irregular nature of the velocity distribution shown in Fig. 7, the overtopping rate is predicted reasonably well using the simple exponential curve. Finally, Fig. 8 also shows that the exceedance probability for the normalized overtopping volume, Q/Qs, can be reasonably well predicted using the simple exponential curve as well.
716
D.T. Cox, C.P. Scott / Ocean Engineering 28 (2001) 707–721
Fig. 6. Exceedance probability for normalized maximum water level on deck with measurements on deck (open circle), measurements at Gage 1 with conditional sampling (solid dot), and predictions using Eq. (4) with a=2.0 (solid), 1.5 (dotted), and 1.0 (dash–dot).
D.T. Cox, C.P. Scott / Ocean Engineering 28 (2001) 707–721
717
Table 3 Comparison of overtopping statistics with conditional crest statistics at Gage 1 Case
Hdh (cm)
N1
(hc⫺Hdh)s (cm)
Hdc (cm)
N2
(hc⫺Hdc)s (cm)
No
(hdm)s (cm)
1 2 3
4.5 6.0 8.5
613 270 52
3.1 2.8 2.7
3.5 5.0 7.5
1070 466 111
3.3 3.0 2.8
469 252 64
4.0 3.8 3.1
4. Summary This study focused on detailed laboratory measurements of the instantaneous free surface elevation in front of a fixed deck and the instantaneous free surface elevation, velocity, and overtopping rate at the leading edge of the deck. Three deck clearances were used which would correspond to the lower decks of a platform. A large number of incident waves (3104⬍N⬍3378) and overtopping events (64⬍No⬍469) were used to increase the statistical reliability of the observations. The tests were conducted using a number of statistically similar shorter runs to minimize effects of tank seiching and re-reflection, and the deck was designed to minimize flow disturbances to the overtopping wave. This study showed that the exceedance probability distribution for the normalized wave overtopping rates and overtopping volumes can be predicted by a simple exponential curve, p(c)=exp(⫺2c). The exceedance probability for the normalized maximum water elevation on the deck could be predicted by the curve, p(c)=exp(⫺2c1.5) although there is no theoretical justification for this expression. This normalized probability distribution is not affected by changes in the deck clearance for the range of deck clearances in this study. The exceedance probability for the normalized maximum horizontal velocity on the deck did not follow any consistent trend for all three cases, although it appeared to follow a Rayleigh distribution of p(c)=exp(⫺2c2) for Cases 2 and 3, except for the extreme velocities measured in Case 2. Seaward of the deck, the nonlinear model of Kriebel and Dawson (1993) gave good agreement to the exceedance probability distribution for the wave crests. Conditional sampling of the crest heights seaward of the deck gave a normalized probability distribution similar to that of the maximum water level measured on the deck for each overtopping event. The values used to normalize each distribution were not the same, however, probably because of the interaction with the structure. Additional studies would be necessary to relate the significant overtopping statistics including (hdm)s, (um)s, (qm)s, and Qs to the incident wave conditions and deck height. Acknowledgements This work was sponsored by the National Science Foundation Career Award CTS9734109 and by the National Science Foundation Research Experience for Under-
718
D.T. Cox, C.P. Scott / Ocean Engineering 28 (2001) 707–721
Fig. 7. Exceedance probability for normalized maximum horizontal velocity on deck with measurements (open circle) and predictions using Eq. (4) with a=2.0 (solid), 1.5 (dotted), and 1.0 (dash–dot).
D.T. Cox, C.P. Scott / Ocean Engineering 28 (2001) 707–721
719
Fig. 8. Exceedance probability for normalized maximum instantaneous overtopping rate, qm/(qm)s (open circle); normalized volume of overtopping water, Q/Qs (solid dot); and predictions using Eq. (4) with a=1.0 (dash–dot).
720
D.T. Cox, C.P. Scott / Ocean Engineering 28 (2001) 707–721
graduates Program. The authors thank Jose Roesset of the Offshore Technology Research Center at Texas A&M University for the use of the laser-Doppler velocimeter, John Reed for technical support in conducting the experiments, and Dave Kriebel for valuable comments on this work.
Appendix A. Overtopping repeatability and sensitivity To assess the repeatability of the measured overtopping, Run 1 of Case 1 was repeated 10 times. Fig. 9 shows the ensemble average of the instantaneous overtopping rate, qa, in the top panel. The middle panel shows the detail in the range 109⬍t⬍116 s with qa indicated by a solid line and the envelope of one standard deviation indicated by the dash–dot line. Although the deviation of the overtopping (±50%) is much larger than the free surface elevation measured at Gage 1 (±2%) (figure not shown), it was acceptable for this study since estimates of the instantaneous overtopping rate can typically vary by a factor of 2. The bottom panel shows
Fig. 9. Measured overtopping for Case 1. Top panel: ensemble average of 10 runs. Middle panel: detail of top panel in the range 109⬍t⬍116 s with standard deviation envelope. Lower panel: instantaneous overtopping for Run 1 repeated with u measured at z=2.0 mm (solid), z=4.0 mm (dash–dot), z=8.0 mm (dashed).
D.T. Cox, C.P. Scott / Ocean Engineering 28 (2001) 707–721
721
the sensitivity of the instantaneous overtopping to the elevation of the LDV above the deck. Run 1 of Case 1 was repeated three times with LDV positioned at z=2, 4 and 8 mm above the deck. The figure shows that q generally increases with LDV elevation above the bed. However, given the scatter for a single run (shown in the middle panel) and given that the exceedance probability curves were generated using normalized values, the conclusions of this study are not sensitive to the LDV position.
References Bea, R.G., Xu, T., Stear, J., Ramos, R., 1999. Wave forces on decks of offshore platforms. J. Waterway, Port, Coastal, and Ocean Engineering 125 (3), 136–144. Forristall, G.Z., 1978. On the statistical distribution of wave heights in a storm. J. Geophysical Research 83 (C5), 2353–2358. Forristall, G.Z., 1997. Wave crest distributions: observations and second order theory. In: Proceedings Ocean Wave Kinematics, Dynamics and Loads on Structures. American Society of Civil Engineers, pp. 372–395. Franco, C., Franco, L., 1999. Overtopping formulas for caisson breakwaters with nonbreaking 3D waves. J. Waterway, Port, Coastal, and Ocean Engineering 125 (2), 98–108. Goda, Y., 1985. Random Seas and the Design of Maritime Structures. University of Tokyo Press, Tokyo. Kriebel, D.L., Dawson, T.H., 1993. Nonlinearity in wave crest statistics. In: Proceedings Ocean Wave Measurement and Analysis. American Society of Civil Engineers, pp. 61–75.