Amplification of waves by a concrete gravity substructure: Linear diffraction analysis and estimating the extreme crest height

Date August 2006

Author

E J van Iperen G Z Forristall, J A Batjes & J A Pinkster

Deift University ofTêchnology

Ship Hydromechanics Laboratory

Mekelweg 2, 26282 CD Delft

TUDeift

D&ft Uflivérsity Of Technology

Amplification of Waves by'a Concreta Gravity

Substructure: Linear DiffractiOn AnälysiS and

Estimating the Extreme Crest Height

by

E.). van Iperen, G.Z. FOrristall,, ).A. Batjes&

J.A. Plnkster

Report No

1494-P'

2006

Publication: Journal of Offshore Mechanics and Arctic

E. J. van Iperen

e-mail: Erik.vanlpere'n@the!Lcorn

G. Z'.. FOrristall

,e-mail. g.lorristall@shell.com:

Shell lnternatlònáFExploration and Productlon P.O. Box 60, 2280 AB Rijswijk, The Netherlands

J. A. Battjes

'Dèpartment of Civil Engineering. TU-Delit, RO. Box 5048, 2600 GA DelIt, The Netherlands e-malt: j.bftjes@ctludeIlt:nl

J. A. Pinkster

ShipHydromechanics Laboratory, TU-Delit, Mekelweg 2, 2628 CD Detti The Netherlands e-mail: j.a.pinksler©wbml.ludelli.nl

Purpose and Scope

A concrete gravity sUbstructure (CGS) as support structure for topside facilities is one of many concepts for an offshore produc-tion platform. A CGS usually consists of a box-shaped base sup-porting large vertical columns. A steel deck with processing equipment is installed on top of the columns.

As a result of the large size of the structure, the incident waves change as they pass through and over the structure. At present physical model tests are required to investigate wave interaction with the structure. Due to the high costs for such tests, they are performed only for the final design. The objective of these tests Is to determine the deck elevation required to avoid green-water im-pact under extreme storm conditions. The deck elevation is of great importance to the overall design of the platform. It is there-fore veiy valuable toobtain an estimate for the required elevation, for various alternatives in an early stage of the design process, prior to the choice for a final design. Eliminating the need: for physical model tests' would imply a significant cost reduction. At present physicalmodel tests are howeverthe only tool available to establish the required deck elevation for a CGS with sUfficient

'accuracy.

'The purpose of the present study is to investigate the féasibility of numerical simulation of the wave enhancement over the COS based on linear' diffraction theory, supplemented with local non-Iinear:corrections for the estimation of extremecrest 'heights. The

Contributed by:theOcean:Offshore and Arctic EngineennglDjvjsjon[of'AsME:for publicationhinthe JOURNAJ.0FOflSHOREMECHANICS AND ARenc ENGINEERING. Manu script receIved August 24, 2004; final manuscript receivedh July 7. 2OO5 Assoc. Editor: Dan Valentine:,

Amplification o.f Waves by

a

Concrete Gravity SubstructUre:

Linea r

_{if fra:c'tî on "Ana lysis}

and Estimating

_{the Etreme}

Crest Height

D/fraction of both regular and irregular waves by a concrete gravity substructure (CGS)

was investigated using experimental surface elevation data and Computational results of the linear diffraction code DELFRAC. The influence of ¡he box-shaped base that supports

the four vertical cOlumns was studied independently from the columns, using data from

regular wave model tests of the Malampaya CGS. DELFR..4C was shown to give accurate results for the focusing of waves over the submerged structure. Results from regular wave

data analysis of model tests of the complete Sakhalin II project Lunskoye CGS were

compared to the predictions by the linear dj/frac:ion code. For the wavecases tested, the

first-order amplitudes were accurately predicted. Dffraction of irregular waves at the

Lunskoye CGS was studied in asimilar way and linear djffraction theomyforrandomseas

gave an excellent prediction of incident wave spectral dy7raction, including the peaks in

the diffracted'spectruni near twice ¡he peak frequency in ¡he inputspec:rwn. The results obtained forthe Lunksoye CGS in ¡he present study wereconsistent with resultsfound in

similar studies on less complex structures. An attempt to predict the extremecrest heights

from the djffracted spectrum was made using a Weibull distribution, and _{a second-order}

expansion of the sea surface that captures the effects of wave steepness, water depth, and directional spreading with no other approximation than the truncation of the expansionat

second ordec Depth indu ced breaking appeared to be an important phenomenon limiting

the crest heights. The crest heights in a 100-year sea state at the Lunskoye CGS_were accurately predic:ed [DO!: 10.1115/1.2199562]

linear diffraction code DELFRACis used to predict the amplifica-tion of waves by the structure. We first look at the accuracy of

DELFRAC_{in predicting the interaction with the box-shaped}

under-water base alone. This analysis concerns regular waves only. We then continue with regular waves interacting with a complete CGS. In the last step of the diffraction analysis, we proceed with inegular waves interacting with a complete CGS. For a given random sea-state, the diffracted spectra are predicted at various

locations around the structure and compared to the measurements.

Various options to derive a maximum crest height from a dif' fracted spectrum are explored. We assess the applicability of the Rayleigh distribution, followed by a similar analysis for the Weibull distribution. The next option compares a second-order expansion of the sea surface to the measurements of the highest crests. Depth induced breaking is identified as a possible explana-tion for the loss of energy in the diffracted spectrum, which sig-nificantly influences the extreme crest height distribution.,

'The Available Datasets

In this paper we use results from two model studies. The spe-cific tests that are analyzed and their most important parameters are briefly discussed in this section. All valùes are at prototype

scale.

The Malampaya ModeL Study. The 'tests analysed consistedof regUlar, unidirectional waves in a wave flume. The waves passed overa submerged caisson of two diffèrent geometries, approach-' ing the structure head: on We:used! the measurements with only

the:base caisson and noi legs present: These measurements

supple-metited the tests of the Sakhalin structures, for which only the

Journal of Offshore Mechanics and Arctic Engineerlñg, _{AUßUST 2006, Vol; 128 1 21Ìt}

complete CGS was modeled. The water depth was 45 m. The

parameters of the wave cases are shown in Table 1, the geometries

are shown in Table 2.

At several locations, surface elevation measurements were made of both the undisturbed waves, and of the waves interacting with the caisson. Results from these tests are analyzed and com-pared to predictions made by DELFRAC. This allows us to assess the accuracy of the numerical code for wave interaction with the

base.

The Sakbalin H Model Study. The Sakhalin II project com-prises two offshore production platforms, one at the Lunskoye field, and another one at the Piltun-Astokhskoye field. We will refer to these structures as LUN-A and PA-B. The complete CGS structures were tested under various wave conditions. In this pa-per, we analyze a number of tests of the LUN-A CUS. The results

are compared to predictions made by DELFRAC. Only one test of

the PA-B CUS is analyzed here. In all cases the centerline of the structure was at zero degrees to the normal to the wave genera-tors; the waves approached the structure head on.

LUN-A CGS. The water depth during the LUN-A tests was 53 m. Additional parameters of the wave cases analysed are

shown in Table 3.

The geometry of the LUN-A CUS, as shown in Fig. 2, is

roughly

Table i Malampaya wave parameters

No. T (sec) _{H (m)} L (m) _[-] Ur [-]

Fig. i The Malampaya CGS

Table 2 Matampaya caisson geometries

Table 3 LUN-A wave parameters

Fig. 2 The LUN-A CGS model

Base

Length: 120 m forward to aft Width: 110 m port to starboard Height: 15 m

Concrete shafts

Spacing: 68 m forward to aft 40 m side to side Height: 45 m top to bottom Diameter: 26 m at the bottom

24 m above midheight

The box-shaped base or caisson, the large diameter concrete columns, and the small diameter steel columns that support the deck are easily identified. The slender black cylinders at the left-most leg are the conductors. They have no structural purpose but transport hydrocarbons up to the deck where they are processed.

PA-B CGS. The water depth during the PA-B tests was 34 m.

The additional parameters of the wave case analyzed are shown in Table 4.

The geometry of the PA-B concrete substructure is roughly:

Base

Length: 115 m forward to aft Width: 110 m port to starboard Height: 14 m

Concrete shafts

Spacing: 34 m forward side to side 56 m aft side to side Height: 27 m top to bottom Diameter: 24 m forward

22 m aft

Table 4 PA-B wave parameters

TR H,ll,

No. Wave type (year) (m)

T,T y o(0)

(secS [_] (o)

212 I Vol. 128, AUGUST 2006 _{TransactIons of the ASME}

No. Wave type

TR (year) H,H,, (m) T,T (sec5 y [_J o(0) (°) I Regular 10 12.7 2 Regular 14 12.7 3 3D Irregular 100 9.9 14.3 2 15 4 2D Irregular loo 9.9 14.3 2 -5 3D Irregular 10,000 13.8 17 2 15 13.4 17.4 264 0.07 13 2 10.8 16.4 179 0.09 5.8 3 9.0 13.0 126 0.10 2.3

Caisson Length (m) Width (m) Height (m)

A 115 lOO 19

B 90 75 16

FIg. 3 The PA-B CGS model

Linear Diffraction Model Delfrac

The linear diffraction codeDELFRACwas developed by Pinkster

[I]. Dmitrieva [2] described the program and compared calcula-tions and experimental data. Some of this description is

repro-duced in this section.

DELFRAC_{is a radiation/diffraction program, which was}

devel-oped for the analysis of the interaction of surface waves with

ships and offshore structures. lt solves the linear velocity potential

problem using a three-dimensional source distribution technique. The program calculates the wave loads and motion responses of a

free-floating or fixed structure in regular waves and is applicable to shallow as well as to deep water.

The fluid is assumed to be nonviscid, homogeneous, and in-compressible, and the flow irrotational. The radiation and diffrac-tion velocity potentials on the body's wetted surface are deter-mined from the solution of an integral equation obtained using Green's theorem with the free-surface source potential as the Green's function. For the computations, the mean wetted part of the object is approximated by a number of quadrilateral or trian-gular plane elements, representing a distribution of source

singu-1.4 1.3 1.2 1.1

fi

0.9 0.9 10

o

0.9 0.8 10

o

lanties. Each of these contributes to the velocity potential describ-ing the fluid flow. Knowledge of the velocity potential around the

structure is sufficient to compute the fluid pressures and wave

loads.

In the present research, the DELFRAC_{output yields the wave} elevation amplitudes and phase angles. These are given at several locations around the structure, including those of the water level probes during the Sakhalin II project model tests.

Regular Wave Diffraction Due to the Base of a CGS

To assess the accuracy ofDELFRAC_{in predicting the diffraction}

of regular waves interacting with the box-shaped underwater base

alone, its output is compared to the experimental results from the Malampaya model study. An array of measurement probes was

deployed along the centerline of the structure. Measurements of the undisturbed waves without the structure in place were also made. Comparison of the time series of the disturbed and the undisturbed tests yields information about the amplification of

waves due to the structure. Because the test data was not available digitally, a separation into harmonic components was not possible. We therefore simply plot the amplified wave height normalized by

the incident wave height. Figure 4 shows the experimental and computational results for the three regular wave cases interacting with the large caisson. The results for the smaller caisson are

shown in Fig. 5.

At the upstream end of the structure during test I, the increase in wave height is immediately apparent. lt is not reproduced by DELFRAC_{but is relatively small compared to the values recorded} elsewhere. However, as the wave profile both adjusts to the local change in water depth and undergoes the effects of wave

diffrac-tion, the increase in wave height is considerably enhanced. As the waves propagate over the underwater caisson, their phase velocity

reduces. This leads to a curved wave front, which in turn causes focusing of wave energy toward the rear of the structure. There-fore the largest effect will clearly be generated along the center-line of the structure.

Comparison of the results from the different tests suggests that the interaction with the base is strongly dependent on the wave-length of the incident waves. lt can be argued that as the incident wavelength reduces, the effective reduction in water depth

be-Te. I

20

o

T. 3

30 40 50 60 10 80

Location along cant,ellne (m)

FIg. 4 Wave Interaction with caisson A

o Maasuremenla

- OELFRAC

90 100 110

Journal of Offshore Mechanics and Arctic Engineering _{AUGUST 2006, VoI. 128 I 213}

0.9

-0810 _{20 30}

40 60 10 80 90 100 110

Te2

1.4 '1.3' C' O Môeuremente - DLFRAC , J

comes smaller. As a result, the amplification factor might be ex-pected to be smaller. However, wave focusing, the process

de-scribed in. the paragraph above, acts to counter this effect since its magnitude depends on the size of the caisson relative to the

inci-dent wavelength. As a result., significant increases are again ob-served on the downstream side of the structure. Van Iperen [3] made a simple calcûlation showing that shoaling is not the

gov-erning process.

For a complete 'CGS. a box-shaped base with fòur columns

mounted. on top, two separate wave-structure interaction mecha-nisms were clearly identified by Swan {4]. The first was related to

the prupagation of waves over the base Based on the presented

comparison of the Malampaya measurements with the DELPRAC output, we conclude that DELPRAC gives good] results for this

so-called caisson effect or wave focusing The second mechanism

was a separate interaction between the incident waves and the legs

of the CGS. Swan [4] shoWed the two interactions were not

lin-early additive.,

Regular Wave Diffraction Due to a Complete CGS

An array of measurement probes was also deployed along the centerline of the structure during the LUN-A model study.

We analyzed the digital test signals and separated them into

their 'harmoniô components. The amplified first-order amplitude is normalized by the undisturbed: first-order amplitude. To 'be able to

compare the measurements to the results from. linear diffraction theory, the complete model: was loaded, in DELFRAC FigUre 6 shows the measured first-order amplitude amplificatiOn, and the prediction computed' withDELFRAC.

With some notable exceptions, the experimentáli and: computa-tional results' at first order are in close' agreement with eachother,

and with the results found by OhI et al. [5] for a less complex.

structure There. is little difference between the tWo wave-steepness cases, as expected from linear theoiy. The centerline.

plots for both the theory and. the experiment indicatea large peak. approximately attheimodebcenter, which is at roughl'y6O 'maloñg the centerline in' Fig., 6.

Although the: agreement between experimental 'data 'and theory'

is quite close, there aresome discernible 'differences.. The most

2i4 ii

'óLjt28,/UGUST2OO6

Tedi

Fig. 5 Waveinteraction wIth caisson B

important discrepancy is that the simulated minimum in .between

the two, front legs, approximately between x=20 m and x=4O:m

seems to be shifted up Wave of the measured minimum. This. was

also found by 'Ohi et aI [5]. An explanation for this difference is not known. In addition, the amplification factor for the

measure-ments continues to drop after x= 100 m, whereas DELFRAC pre-dicts a slight increase in amplification factor. A possible

explana-.tion for 'this is dissipation of energy due to breaking, which

obviously is not taken into account 'by DELFRAC.

Van Iperen [3] also normalized and' plotted ¡he second- and' third-order harmonics along the centerline. Trends in the

amplifi-cation profile along the centerline were' similar 'to those fòund.'by

Ohi' et al. [S]. The location of peaks and troughs were related]:tó'

the first-order 'profile.

Irregular Wave Diffraction 'Due to a Complete CGS

In the previous. sectiOn, the amplification factors of the mea-sured, first-order harmonic component were derived for' regular

Waves, along the centerline and compared] to. the amplification. fàc-tor .as predicted by DELFRAC. A similar comparison can be made.

for .the irregular waves We define

(x,y).= Sa(x,y,fn)If= m n

the average' of the squared amplified surface elevation

measure-ments with the structure in place, or .the zeroth-moment of the measured] amplified wave variance density spectrum Sa is the variance density of the amplified waves at 'lócatiofl.(x,y) at. fre-quency j,, and] f is the' frefre-quency step size.

S(f)if=m0

the average squared] incident sürface elevation, or the. zemth:mo,.

ment of' the iflcideñt wavevariance density .spectnim S is the variancedensity of the iñcidánt waves at frequency.

(x,y)=

_{xy)Sf,)'tf=moD}

TraniactloflsoftheASME

0 10' 20 30 40 50. Tod 3 80 70' 80 go 20 30 40' 50 60

Location along centrellne (m)

70 80 90

I

as 2.5

i5

0.5 o O

;

1.5-+

X + H1Om X II'14m DELFMC 1

the average squared amplified surface elevation as predicted by

DELPRAC,or the zeroth moment of the variance density spectrum

as calculated by DELIRAC., (X,y)is the amplification factor

cal-culated by DELPRAC.

We also define: an average amplification factor that allows fòr overall comparison of theDELFRACresults to ¡he measurements

For the measurements amplification factor =

Regular wa tests WN.A

btegular wa test 3 LUN.A

ForDELFRAC: amplification factor =

m01

The results are plotted in Fig. 7, whichshows thatDELIRAC is in

very good agreement with the measurements.

The preceding work focuses on the amplification of wave heights along the centerline of the stnjcture.DRLFR Cgives goodE

results for both regular and irregular waves. Wenow considér the

4O 60) 80 100 120E

Location el gcentrallne

Flg 7: AmpIIflóatlOn factorforirregùlarwaves

JòürflalofOffshöio Mecháfllös and Arctic Eflglneerlñg _{AiJêUST2ôo6 VÒIi 128 1 215}

20! 40 80 80

Location along centreilne (m)

FIg. 6 _{Regular waveinteraction with LUN-A CGS}

1001 120

OES

O Measurements

J

600 40d 300 200 100

-

Uflietuibed spectn ru --- Measured d1asted spectrum

DELFRAC dilfracted spectrum

Hno:mea.uraclt4gm

IImOIDELPRAc'.t4.6iw

0.05 0.1 0.15

Frequency(Hz)

detáils of the amplification of the wave spectrum

Figure 8 shows the undisturbed, measured, and diffracted spec-trancar arear leg, where the amplificationof the wave height was greatest. The measured spectrum in Fig. 8 clearly shows an in-crease inenergy at approximately tWice the peak frequency of the

incident spectrum. This is not due to.second-order effects, because DELFRAC, which is a linear diffraction code, also quite accurately

predicts this increase.. This is an indication that the actual diffrac-lion process at the rear legs is predominantly linear. The strongest argument supporting this idea can be drawn from comparison of

lJnElaturbed tregutar wavelests

1pt69 seC tFtmO measUredI3Jm

Tes! 3 rlanóe:derisfty spectrum forpröbe R25

Flg.8 Spectra at right rear leg

-

That 3 Teat 5

the two irregular wave tests, which have idènticali undisturbed high frequency tails as shown in Fig. 9.

Even though the total incident energy of the two sea states is

different, the bumps in the diffracted spectra are very similar. This is consistent with linear diffraction theory, Which impliés:there is

no transfer of energy from one frequency component to another. Inspection of spatial surface elevation plots showed that 'this am-plification at higher frequencies is due to frequency-dependent constructive interference of waves interacting with the colùmns.

1200 t000

F00

i

> 400 200' 0.2 0.28 bTeda,waavteala R25 Tp.l77eec lm0'meaaured-203m Tp'.I45eec tHm0me55d49m

- Test3

-. Teat5 03 n 0.1 0.2 03 0 :01 _0,t 03 IErerpJencyQlz) Piequency (Hz)

FIg. 9 DIffracted spectra for two irregulór sea-states

250

200

g

J

100

Test 3 w1aoce density spectrum

1l4i sec HñiO.9 Sin

0.1 0.2 ₀₃

Freiency tHz)

From Spectrum to Probability Distribution

For the study of maximum green water surface elevation, the interest is flot so much in the wave height or spectrum, but more in the crest elevation.

Rayleigh Dlstribution To first order, crest heights i,, follow the Rayleigh distribution, whichis givenby

i)=exP[_ 8-Ç}

Figure 10 shows that the undisturbed crest heights are not well described by the Rayleigh distribution. For the 100-year multidi-rectional undisturbed sea state, the probability distribution of the crest heights and the troughs are compared to a Gaussian surface in the upper right corner. The lower right corner shows the nor-malized crest heights against the probability of exceedance. The crest heights are normalized by the crest heights of a Gaussian surface at that probability of exceedance. The Rayleigh distribu-tion consistently under predicts the crest height for low probabili-ties. Therefore the crests are higher than the troughs are deep, as expected for non-linear waves.

Weibull Distributiom The Rayleigh distribution has only one variable parameter that can be used to scale the distribution. ¡t cannot be adjusted to fit a distribution with a different shape. The Wèibull distribution, of which the Rayleigh distribution is a spe-cial case, does have an extra variable parameter

P(ii>

aiim0

In Fig., I1, the crest heights are normalized by the crest heights obeying a WeibUll distribution at that probability of exceedance. The parameters of that Weibdll distribution are calculated using, maximum likelihood estimates. The maximum crest heights dur-ing tests 4 andl5 are overestimated by the Weibüll distribution by more than 10% For this parametrization to be useful in predic-tion, 'the Weibull parameters must be related to independènt mea-sures ,ot9the waves, suëh as the steepness and Ursell number

Uñ-J. Io.' io;3 -, Crests, Gaussian Troughs .1 05 IT

'Crest lielaht I limO

Olstdbution,ot creatheights

t0-Probability óf Exceedanca

Fig. 10 UndIsturbed spectrUm and probabilities compared to the

Rayielgh.distrl-bution for test 3

fortunately the Sakhalin tests do not cover enough different valUes

of steepness or Ursell number to result in a robust

parameteriza-tion

Surface Elevation Simulations to Second Order. 'Fonistail [6] numerically implemented second-order wavewave interaction calculations. A secondorder expansión of the sea surfacecan cap-ture the effects of wave steepness, water depth, and directional spreading with no approximation (within the potential-flow

theory) other than the truncation of the expansion at second order. Higher-order interactions and other effects will of course influence

the distribution of real wave crests In panicular, wave breaking could be important. However, the point of the investigation by

Forristall [61 was to see how well' a straightforward' applicatiònof

secondorder theory could match observations during heavy storms: In the present study, the simulation program is used to' answer that same question for the measurements obtained during

the Sakhalin Ii project model tests. The spectrum is first simUlated

using linear wave components. Then the time series isadjusted to include two second-order effects:

Sharpening of the crest and flattening of the trough due to superharmonics, or sum iñteraction terms

Set-up or -down caused by súbharmonics, or difference

in-teraction terms

'For brevity these are referred to as plus and minus terms, respec-tively, in the figures. Simulations were performed both with and without the subharmonics to see which effects were Important Many simulations using different random phases in the linear waves are combined to give a robust sample distribution' of the

second order waves.

Figures 12 and 13 show both the measurements and the com-putational results for the undisturbed conditions

Figure 12 shows that fòr the multidirectiònal sea state,,'tlie,flt between the 'measurements and the siìnulationis;best whenithe

subharmonics areomittedL Fór'the unidirectional sea state,)Fig. 13

showsthe fit tobe'best when all the terms of;the simulation are

includedL

Journal.ofO!fShoreMecharIcsand Arctic EngIneering

ÄUG@ST2006, 'Vo'I. 128 i 217

10°

We have observed similar behavior in other measurements of

undisturbed waves in tanks. Field measurements generally agree

better with simülatiöns iücluding all of the second-order terms. A

possible explanation could be that the subharmonics, i.e., the long

set-up and set-down waves, which are bound to the shorter free

surface waves, arenot absorbed properlyby the wave absorbers in

the wave basin. Oñce the free waves reach the end of the basin, the short waves are bsorbed and the longer, bound waves are reflected into the basin to propagate as free waves and interact

0.0

. 0.6

1.4

0.4

0;2

Otstdbuùon oÍ unietuthed creatheIts

Teat Sc0atflbutlon of Undlaturbed creatheighta

ctp.00Q0o0ooO0 Q, O

+ 'Teat3 O Teat4 TeatS

- -. Welbull dlatdbùtlon

Fig. 11 _{Undlsturbedprobabiiitieg compared totheWelbUil distribution for tests 3,,}

4 and 5

with the other free waves in the basin; The bound set-up or -dòwn terms tend to 'lower the surface elevation under large wave groups. Due to the chaos of interacting long waves, this phenomenon does not develop in the laboratory as it would in the field. The absence of the bound long waves results in higher crest elevation

measUre-ments than predicted by a second-order simulation including

them.

We now continue the analysis with the structure in pläce The

focus is on the right rear leg, because this is the location with the

o Meaxuremenla - - - _{Gauaaian eutface} - 3D-SimulatIon 3D-SimulatIOn Pluatermaionly

I..

Probabirdyof Eicàèedùrice.

- Fig 12 Test3undÍstUrbed meastiÑmeflts and secondorder:simuiation

21,8 I VoL 128,ÄY.GUST2006,

_{TransactionsòftheSME}

l0_,

loo _IO'

Probability of Exceedance

J 0.3

i

0.8 0;4 o Measurements - Gaussian surface - 2DSimula1Ion

2Dlmulation plus talma only

j

1.4

1.2

0.2

highest significant wave height Due to the interaction of waves

with the structure, the mean water level is raised above still water level (SWL). To assessthe influence of this phenomenon, which is

not predicted by the simulation, we plot the normalized crest heights with respect to SWL and mean water level (MWL). Al-though in reality the directionality of the waves will be altered by the structure, we used the undisturbed directionality in the simu-lations. Figures 14 and 15 show the fit between the simulations

and the measurement is consistent with the undisturbed tests: For themultidirectional sea state the fit is best when thesubharmonics

1.6- 1.412 -0.8 0.0 0.4 0:2 10°

Test4 dabibuilon olundlsiu,bed creatheights

Test 3 prabability distribution of normaliaed crest heights (or R25

Pg. 14 Test 3 measurements and simulation loo

Fig. 13 Test 4 undisturbed measurements and second-ordersimuiation

are omitted. forthe unidirectional sea state the fit is best when all

the second-order terms are included

Figure 16 shows theresults for the higher waves inthe

10,000-year multidirectional sea state. At relatively high probabilities, the trend of an increase in the normalized measured crestheight with

decreasing probability is bent downward to a decreasing normal-ized crest height. For the lower sea states in Figs. 14 and 15, a

similarprocess starts at significantly lower probabilities. The mea-surements of the. 10,000-year waves at the rear legs of the

struc-ture are not well described by the simulations. It is likely that the

,...

_a:.-.

..

o MeasurementsSWI-0 Meaaurernent# mean .WL-0 Gaussian surface - 30-SImulatIon

3D-Sirneietlan - tanna ordy

10I l02

Probability of Exceadance

Journal of Offshore Mechanics and.Arctic Engineering AUGUST 2006, Vol. 128 I 219

I0l 10'

I°I

drop in normalized crest height measurements is due to breaking of the largest waves.

Depth Induced Breaking

The breaking of waves has an important limiting influence on the crest height distribution. Neither the linear diffraction code

DELFRAC, nor the second-order simulations take wave breaking into account. Video shots of the Sakhalin H project model tests showed waves were breaking along the centerline of the structure during at least some parts of The model tests. In addition, someof the waves overtop the concrete columns, a process which cannot

1.61.4

-

1.2-TesI4_{prthsbiiity fst,buIlon}ofnormailsed crest heightsfer R25

Test5pfthabilitydlatiibullonofnonnsilsed crestheightsfor R25

be described by a linear diffraction model. We suspect, however, that wave breaking is the key reason that the prediction recipe overestimates the crest heights at certain locations around the structure during certaih model tests. Identifying the breaking phe-nomenon andquantifying the extent of breaking relative to certain parameters could greatly increase the 'applicability of the theories used within this study.

Van Iperen [3] presented an overview of literature on wave breaking and the applicability of the breaking criteria was as-sessed Battjes and' Janssen [7] adjusted the Miche criterion to create a shallow water limit different from 0.88

o MeasuzementsSWLaO MeaaurementameanlM0 Gaussian surface - 3D-Simulation

30-Slmuiatlonpius terms only

loo 10' 10.2

PrebabliftyofExceedance

FIg. 16 Test 5measurementsand simulation

220 I Vol. 128f AUGUST 2006 Transactions of the ASME

Fig 15

o MeasurementsSWLß MesaurementemeanWt.O

- - - _{Gaussian surface}

- 2D-Stmuietian

20-Simulation pius terms only 1oI

ProbsbtiltyofExceedance

ibsi dmensIonla oeheme we'.. weww rdatl'.e wete' depth et R25 05F

H

0.4 02 o;iF Fig. 17 leg s kd u Bw. eqet'we H

¿::;;i'

Depth limited wave heights during test 3 at right rear

Hma, 0.88 1 kd

__7_ = --

tanh('.,V-We are aware that this breaking criterion has been developed in the context of a wave energy dissipation model, not for the pre-diction of heights of individual waves. Nevertheless, the criterion is used here as it does at least allow for straightforward compari-son of the various wave conditions. In Figs. 17 and 18 the plots below, the wave height normalized by the water depth is plotted against thewavenumber k, which is:also normalized by the water deptk

The calculation ofthe wave number is by no means

straightfor-ward. The waveperiod can be obtained directly from the

measüre-ments, and can be used to calculate the deep-water wavelength.

We define theiundisturbed shallow-water wavelengthas the length

of waves at a uniform depth equal to that above the base. Typi-cally the undisturbed shallow-water wavelength is in the order of

several hundreds of meters. For a wave with a period equal to the

peak period of the 100-year incident spectrum, the undisturbed shallow water wavelength is approximately 270 m. However, the

wave profile requires time to adjùst to the suddenchange in water depth; its wavelength does not change instantaneously as soon as

the wave reaches the structure. In addition the legs influence the celerity of the Waves. Cross-correlation analysis between

mea-surements made atan up-Wave añd a down-waveprobe along the

centerline could provide the celerity of:the crest and thereby the wavelength. The complicated modification of the waves by dif-fraction woûld however make this exercise of questionable

valid-ity in determining the wavelength of the waves over the structure.

Because of the ambiguity inthewavelength, we triedthreediffer-ent options for specifying it. For option 1, the-plotsare made as if the water depth is still the undisturbed water depth of 53 m. For option 2 the disturbed water depth of 38 m is used to normalize and to calculate the wave number as if the waves were at this

water depth traveling over a graduali slope. Finally for option 3 the

wave number is calculated as if the water depth was stili the

undisturbed water depth of 53 m but the water depth used to nor-malize is the disturbed water depth of 38 m. In all calculations we

used y=O.833. Figure 17 shows the wave heights so obtained compared to the breaking curve at the right rear leg for the 100-year waves, and Fig. 18 shows the wave heights for the larger

104000-year conditions.

The figures shoW that the influence of the water depth used for

normalization is very large. Option I uses 53 m and is signifi-cantly different than options 2 and 3, which use 38 m. The water

Teet 5 dknwwloetee. extwme wa%e Ietgflt eewea etett,. wate. dejh el R25

0.91 _ul 0 o

Ho

H

I2

-l'o-Is

L '.9°w H .-... V..

.. ...--..

. t

i 1_

9 9- -lO

FIg. 18 Depth limited wave heights durIng-test 5 at right rear

leg

depth used to calculate the-wave number is of less influence as-can beconcluded from the relatively small difference between options

2 and 3. It appears that for waves with similarperiods;.and

there-fore deep-water wavelengths, the reduction in wavelength -is not

much different -for the two water depths. The figure is consistent

with linear theory that states the difference is larger for low wave-numbers, or long periods.

The Battjes- and Janssen breaking criterion does not provide a clear upper bound for the-wave heightsin either test..Furthermore, more of the waves exceed- the criteria when the input waves-; are higher (Fig. 18). Other breaking criteria described in the literature

did not improve the prediction of the limiting wave height. This-result is not too surprising since the prediction of breaking is difficult even for undisturbed waves4 and it must be even more difficult in the chaotic conditions between the legs of the

struc-ture.

It may be possible that using some modification of a breaking criterion along with the second order simulations could improve the predictionof-the crest height distribution, but that line ofwork

has- not yet been pursued.

Analysis of the PA-B CGS

Van Iperen [3] -also made a limited analysis of the measure-ments of wave interaction with the PA-B structure. The amplifi-cation factors along the centerlines ofLUN-A and PA-B are

plot-ted in Fig. 19. The fit between DELIRAC and the measurements is

much better for-the LUN-Athan for the PA-B structure.Although

DRLPRAC consistently over predicts theamplification factor for the PA-B structure, the global trend along thecenterlineis reproduced

quite well. For the PA-B structure, the maximum value of the measured significant wave height occurs in front of the right up-wave column. The variance density at that location predicted

us-ing DELFRAC (not shown) is significantlyhigher than the measured

value, although the shape of the predicted spectrum is approxi-mately correct. The crest heights are also over predicted, even when the measured spectrum is used as the input to the second

order simulations.

One possible cause of the difference in -accuracy of the -predic-tions for-the two structures is that the front.columns are-consider-ably closer together in the PA-B structure. Reflections from these

columns may cause wave breaking in front of -the CGS, which

reduces the wave energy passing through the structure.

2 0.5 o 2

Ï

+ 222 I Vol. 128, AUGUST 2006 20

Inegular 100-year muItIdrectionaI wa test PA-B

+

40 60 00

Lacadonatong centredne (m)

bTeguIar 100-year multldlrectlonel waw teat LUN-A

+

- Deltac

+ Measurements

Conclusions

The linear diffraction code DELFRAC gives useful predictions of

wave spectra measured inside complicated structures made up of large cylinders. Second-order wave simulations can predict the disturbed wave height distributions until the distributions are modified by wave breaking.

DELFRAC gives a close fit to the measurements madeduring the

Malampaya model test by Swan [4] of regular waves focussing

over the underwater box-shaped storage caisson. lt accurately

pre-dicts the first-order free surface response for the two regular wave cases interacting with the complete LUNA structure during the Sakhalin II project model test.

DELFRAC effectively simulates the shape of the measured

dif-fracted múltidirectional irregular wave spectrafor both theSakha-lin H project structures. For the 100-year sea state at the LUN-A

structure, DELFRAC gives an excellent prediction of the diffracted

spectrum at various locations around the structure, including the peak at twice the incident peak frequency. The results found for the LUN-A structure in the present study are consistent with re-sults found in similar studies by OhI et aL [5] using a different first-order diffraction code on less complex structures. For the

PA-B model the 100-year DELFRAC predictions are too high

corn-paredto the measurements obtained during the Sakhalin H project

model test.

A complete second-order expansion of the sea surface accu-rately predicts the extreme crest height for the model test of the undisturbed unidirectional 100-year irregular sea state at the pro-posed location for the LUN-A structure. Measurements made dur-ing undisturbed multidirectional irregular model tests give higher values for the crest height than expected from second-order

theory.

A complete second-order expansion of the sea surface accu-rately predicts the extreme crest height, measured at the signifi-cant location underneath the deck of the LUN-A structureduring the model test of the unidirectional 100-yearirregular sea state. A second-order expansion of the sea surface excluding the subhar-monics accurately predicts the extreme crest height, measured at

the significant location underneath the deck of the LUN-A struc-ture during the model test of the multidirectional 100-year

irregu-lar sea state.

Depth indùced breaking is an important phenomenon limiting the crest heights.

Acknowledgment

The work performed on this subject as part of the TU-DeIft M.Sc. thesis, and the additional time involved in preparing this

paper, was supported by the Civili Structures, and Marine

depart-ment of Shell EP Projects. Many thanks to Engineering manager Frank Sliggers and his team

Nomenclature

T= wave period

= peak period

TR = return period of the sea-state in years 11= wave height

H5 = significant wave height

Hmax maximum depth limited wave height = _{amplified wave height}

If =

incident wave height

L=

wavelength

5= H1/L=wave steepness

Ur = H1Id(L/d)2=Ursellnumber

xth-order amplified regular wave amplitude

A=

incident first-order wave amplitude

k=

wavenurnber

d=

water depth

7= JONSWAPpeak enhancement factor

/1(0) mean wave direction relative to the normal to

the wave generator

standard deviation of the frequency integrated angular spectral energy distribution. For the

cr(0)

- Deltac

+ Measurements

Transactions of the ASME

20 40 80 50 loO 120

Location along cenlreilne Cm)

Fig. 19 Comparison of DELFRAC fit to measurements forPA-B and LUN-A

multi-directional spectra (3D), a(0)= 15. For the unidirectional spectra (2D) the standard

deviation is zero.

References

[I) Pinkster. i. A., 1995, "Hydrodynamic Interaction Effects in Waves," Proceed-ings 5th International Offshore and Polar Engineering conference, The Hague. ISOPE, Golden, Vol. UI, 414-419.

[2] Dmitrieva, I., 1994, "DELFRAC 3-D Potential Theory Including Wave Du-fiaction and DriftForces Acting on the Structures," TU-Delft Report No. 1011, December.

Journal of Offshore Mechanics and Arctic Engineering _{AUGUST 2006, Vol. 128 I 223}

Van Iperen. E. j., 2003. "AmplifIcation of Waveslby a Concrete Gravity Sub-structure," M.Sc. thesis TUDelft.

C. Swan & Associates LUI.. 1998, "Wave-Structure Interaction st tise Proposed Malampaya CGS." Final reporito ShellIntemationsI Esplortion and'Produc-tion; July.

ObI, C. O. G., Estock Taylor. R., Taylor,P. H., and Borthwick; A G. L., 2001, "Water Wave Diffraction by a Cylinder Array. Pars I. Regular Waves," J. Fluid

Mech., 442, l-32; "Water Wave Diffraction by a Cylinder Array. Part 2.

Irregular Waves," 442, 33-66.

Fomstall, G Z., 2000, "Wave crest distributions: Observations and Second-Order Theory." J. Phys. Oceanogr.. 30, 1931-1943.

[7) Battjes, j. A.. andJanssen, J P. F. M., 1978, "Energy Loss and Set-Up Dueto Breaking of Random Waves," Proceedings 16th International Conference on Coastal Engineering, Hamburg, ASCE, New York, pp. 569-587.