wave run up

Marine Structures 16 (2003) 97–134

Comparative study on airgap under floatingplatforms and run-up along platform columns

Finn Gunnar Nielsen*

Norsk Hydro ASA, Hydro E&P Research Centre, Box 7190 5020 Bergen, Norway

Abstract

Present engineering practice for computing airgap (i.e. the clearance between waves and

deck) on floating platforms relies on very simplified approaches. However, recently several

new codes for computation of nonlinear wave diffraction have been developed. To obtain a

state of the art of the capabilities of these methods, test cases related to airgap under a floating

platform and run-up along platform columns have been defined. Several organizations have

been invited to apply their numerical tools to compute airgap and run-up for the defined test

cases. The results from the comparisons are summarized and compared to experimental

results. The study has been part of the work of ISSC2000 Committee 1.2. r 2002 Elsevier

Science Ltd. All rights reserved.

Keywords: Runup; Nonlinear waves; Computational methods; Model tests

1. Introduction

In the design of floating platforms for harsh environment, requirement to calmwater deck clearance is an important consideration. Sufficient deck clearance mustbe ensured to avoid damage in deck due to waves hitting from below. On the otherhand, increased deck clearance is an important cost driver. Accurate assessmentof the necessary deck clearance is, therefore, important. The airgap is defined asthe clearance between a platform deck and the wave crests. The heights of the wavecrests are given from the incident waves with the addition of diffraction–radiationeffects due to the presence of the platform. A jacket platform with small diameters,legs and bracings disturb the incident wave field only marginally while asemi-submersible with large diameter columns and pontoons has significant

*Tel.: +47-55-99-69-10; fax: +47-55-99-66-00.

E-mail address: [email protected] (F.G. Nielsen).

0951-8339/03/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved.

PII: S 0 9 5 1 - 8 3 3 9 ( 0 2 ) 0 0 0 2 3 - 0

diffraction–radiation effects. In addition the platform moves in waves. Theinstantaneous vertical position of the deck must be considered in evaluating theairgap. Run-up is a highly nonlinear phenomenon confined to the near field ofthe platform columns. Radiation-diffraction effects cause the run-up, but theextreme nonlinearity is confined to a region of close proximity to the wall, i.e. adistance much less than one column radius.

Large loads are to be expected if waves hit the platform deck. To quantify theloads, the magnitude of the ‘‘negative airgap’’, as well as fluid velocities are needed,see e.g. Baarholm, Faltinsen and Herfjord [1]. In the present study it is focused onthe deck clearance only.

Several authors have considered the diffraction–radiation and run-up of wavesaround single and multiple circular cylinders. McCamy and Fuchs [2] gave theclassical linear solution of the diffracted wave field around a bottom mountedcircular column. Niedzwecki and Duggal [3] performed small-scale model tests ofrun-up on a bottom mounted circular column. They show that linear diffractiontheory under-predicts the run-up except for very low wave steepness. Second-ordermethods for computing the wave field around single circular cylinders have beengiven by e.g. Kim and Yue [13] and [14]. Second-order methods based uponboundary element techniques and valid for diffraction–radiation of general bodygeometries are described by e.g. Lee and Newman [4]. Sung and Choi [5] and Ferrant[12] describe fully nonlinear methods for the diffraction problem. Kriebel [6] hasmade a simplified approach to compute run-up along cylinders.

Present engineering practice handles airgap and run-up estimation in a very simplisticmanner, e.g. the NORSOK standard [15] states that: ‘‘The wave enhancement andmodification of the kinematics caused by the structure may be estimated by linearradiation and diffraction theory. The fact that the wave crest is higher than given bylinear theory shall be accounted for. Model tests should be performed if theoreticalmethods cannot predict the effect of substructure with sufficient confidence due tononlinearities related to steep waves, and wave–current interactiony.’’

In practice the wave crest elevation is estimated by applying a factor on theestimated extreme value based upon a Rayleigh distribution of maxima, i.e. themaximum crest elevation is estimated from

Xmax ¼ asffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 lnðNÞ

p; ð1Þ

where s is the standard deviation of the wave elevation process including the effect oflinear radiation and diffraction effects, N is the number of waves in the periodconsidered, and a is a factor accounting for asymmetry in the waves. For airgapestimation under semi-submersibles, a-values in the order of 1.3 is found. Forfloating platforms linear and slow drift motions must be accounted for in estimationof the airgap.

As the capabilities of computing airgap and run-up is an important input toslamming load computation, the load committee of ISSC 2000 [7] decided to initiatea comparative study to investigate present capabilities of state-of-the-art numericaltools. Similar comparative studies have been executed before, e.g. the ISSC 1997study on mooring line damping by Brown and Mavrakos [8]. Herfjord and Nielsen

F.G. Nielsen / Marine Structures 16 (2003) 97–13498

[9] made a comparative study on computation of hydrodynamic coefficients foroffshore structures.

In the present study it is investigated how well state-of-the-art linear and nonlinearradiation–diffraction programs are able to compute the airgap and run-up alongplatform columns. The test cases defined are all related to real platforms. Threeseparate geometries, cases A–C have been considered in the study:

(A) A rigid vertical truncated circular column. The column is fixed and exposed tomonochromatic and bichromatic waves.

(B) The second case is identical to case A except for the column geometry. In thiscase the column has a square cross-section with rounded corners.

(C) The third case is a complete semi-submersible with four columns and fourpontoons. The platform is free to respond to the waves. It was exposed toregular waves of different steepness.

For all test cases experimental data exist.About 50 individuals and organizations were invited to submit their calculations.

All have recognized expertise in hydrodynamic modeling. As could be expected, notvery many are able of performing this kind of computations. Some of these had nosuitable tool for such computations, others had no internal funding for the work. Weare, therefore, very grateful to the six organizations who contributed to the study.Not all provided results for all the specified test cases. Others submitted results frommore than one method. Thereby altogether nine computational tools have beenapplied in the study. The analyses were performed during 1999. The results werecompiled and compared to experimental results in early 2000. A summary of theresults were presented at the ISSC 2000 conference [7].

In the following, the cases considered are presented in detail and the resultssummarized. All results are presented graphically. Numerical, tabulated values arenot included, as a comparison to ‘‘graphical accuracy’’ seems to be sufficient. In thepresent comparison, results from mono- and bichromatic waves are included. Somecomputations have been performed in irregular waves as well. These results are notincluded as they are performed by linear computations only.

2. Specification of computations

2.1. General considerations

For cases A and B a water density of r ¼ 1000 kg/m3 is assumed, while in case C,r ¼ 1025 kg/m3. All incident waves are long-crested and propagating in positivex-direction (01). To avoid the problem of asymmetry in the specification of incidentwaves, the incident wave height, H (double amplitude) is specified. For the truncatedcolumn cases (A and B) the water depth is 489 m, while the depth in case C is 325 m.

For the monochromatic waves the wave height and wave period is specified.However, for the bichromatic waves, time series of the incident wave elevation in

F.G. Nielsen / Marine Structures 16 (2003) 97–134 99

origin x ¼ y ¼ 0 were available. Thereby any confusion about phasing of the wavecomponents is avoided. The wave elevation is positive upward with zero at the calmfree surface.

2.2. Case A, circular column

A vertical truncated circular column is considered. The diameter D ¼ 16:0 m, andthe draft of the column is 24.0 m. The column extends vertically above the freesurface with constant cross-section. The wave elevation is to be computed in thepoints as given in Table 1. The radial distance is measured from the center of thecolumn. The positions are illustrated in Fig. 1.

The wave conditions are given in Table 2.

2.3. Case B, square column

The square column has a width of 16.0 m and a draft of 24.0 m. The radius of thecorners is 4.0 m. The width of the plane section on each side is 8 m, see Fig. 1. Thewave elevation is computed in the points as given in Table 3. One side is facing thewaves. The wave conditions are identical to case A.

Table 1

Positions where the wave elevation is computed. Column A

Row Direction (1) Radial distances (m) point no. 1, 2, 3 and 4

A1 270 8.05, 9.47, 12.75, 16.0

A2 225 8.05, 9.47, 12.75, 16.0

A3 202.5 8.05, 9.47, 12.75, 16.0

A4 180 8.05, 9.47, 12.75, 16.0

Fig. 1. The circular (A) and square (B) columns. Positions where the wave elevation is computed are

shown (the illustration has not correct proportions).


2.4. Case C, floating platform

The dimensions of the floating platform are given in Table 4. Fig. 2 illustrates theplatform. In Table 5 the wave conditions used are specified. In Nielsen et al. [10]more information about the platform may be found. The platform motions arecomputed without considering restoring forces from mooring and risers. If somekind of horizontal restoring forces are required for the computations, a constanthorizontal stiffness equal to 3.53� 105 N/m should be used in surge and sway. The

Table 2

Wave conditions considered. Combinations of wave heights and periods

Monochromatic waves Wave height, H (m) Wave period, T (s)

M1 4.22 9

M2 7.9 9

M3 12.65 9

Bichromatic waves H1 (m) T1 (m) H2 (m) T2 (m)

B1 2.55 7 4.22 9

B2 7.9 9 14.05 12

B3 7.9 9 21.96 15

Table 3

Positions where the wave elevation is computed. Case B

Row Direction (1) Radial distances (m) point nos. 1, 2, 3 and 4

B1 270 8.05, 9.47, 12.75, 16.0

B2 225 9.707, 11.127, 14.407, 17.657

B3 180 8.05, 9.47, 12.75, 16.0

Table 4

Particulars of the semi-submersible, Case C

Mass of platforma 189.0� 106 kg

Displacement of platform 187100 m3

Density of water 1025 kg/m3

Vertical force from risers and mooring 27.3� 106 N

Draft 40.0 m

Center of gravity (below water line) 11.8 m

Center to center distance between columns 72.52 m

Diameter of columns 29.0 m

Height of pontoons 14.5 m

Width of pontoons 29.0 m

Radius of gyration (referred to CG) ryy 43.6 m

a Difference between buoyancy and weight is due to vertical forces from mooring lines and risers.


locations where the airgap is computed are given in Table 6. Origin of the coordinatesystem is at the mean water surface, z pointing upwards and x ¼ y ¼ 0 in themidpoint between the columns. The x and y axes are parallel to the pontoons.

3. Computational results

3.1. Participants

Six universities/companies have submitted results to the study, see Acknowl-edgements. None of the contributors analyzed all cases. The individual results arenot identified, only denoted, methods A–H. There are more methods referred than

Fig. 2. Hull of the floating platform, Case C, submerged part.

Table 5

Wave conditions for the semi-submersible, Case C

Monochromatic waves Wave height, H (m) Wave period, T (s)

M11 6.09 10.94

M12 7.55 10.92

M13 11.14 10.92

M14 19.25 10.95


contributors as some contributors have employed more than one method orcomputer program. The contributors were asked to provide information about theirmethods. A short summary of the methods used is given below. Further details aboutthe methods are found in references [15–32].

3.2. Methods used

Table 7 summarizes the replies from a questionnaire submitted to the participants.As can be observed, in most cases boundary element methods (BEM) are employed.

Table 6

Positions where airgap is computed. Case C, semi-submersible

Point No. x-coordinate (m) y-coordinate (m) Comment

1 0 0 Center point

2 36.6 0 Midpoint between aft columns

3 0 36.6 Midpoint between for and aft column

4 18.6 36.6 Front of aft column

5 24 24 Front of aft column

6 �54 36.6 Front of forward column

Table 7

Summary of methods used

A B C D E F G H I

Basic method BEF BER BEF BEF BER BER BER ANA BER

Domain FD TD FD FD TD TD TD FD TD

Perturbation 2nd 2nd 1st 1st 2nd FN FN 1st 2nd

Motion included

(Case C)

NO YES YES YES

Computer DIG PC PC PC Dec HP HP PC C90

CPU time per wave

period (min) Cases

(A:B:C)

5:-:9 80:100:- 0.5:0.5:4 3:3:4 10:10:- 60�120:-

:-

80:-:- -:-:0.3 60:-

:-

Abbreviations

BEF: Boundary element technique, Free surface Green’s function.

BER: Boundary element technique, Rankine Green’s function.

ANA: Semi-analytical technique (interaction between vertical cylinders).

FD: Frequency domain.

TD: Time domain.

2nd: Perturbation to second order in wave steepness.

1st: Solution linear in wave height.

FN: Fully nonlinear free surface condition (single valued).

DIG: Digital workstation.

PC: Personal computer, Pentium processor, 200–450 MHz.

HP: Hewlett-Packard 9000/K210 workstation.

C90: Cray C90 1 processor.


Most of the linear BEM codes use a free surface Green’s function formulation andemploy frequency domain analysis, i.e. a Green’s function satisfying the linear freesurface condition. Singularities are distributed over the body boundary only. Thenonlinear methods, except for one second-order method, all employ Rankine Green’sfunctions and time domain analysis; i.e. singularities are distributed over the bodyboundary as well as the free surface. The second-order methods are based upon asecond-order perturbation scheme of the boundary conditions on the body and thefree surface. The perturbation is performed around the mean position of the surfaces.However, in the present study none of the participants have solved for the second-order radiation effects, which are relevant for case C, i.e. the second-order effects areconfined to the treatment of the free surface boundary condition only. The ‘‘fullynonlinear’’ methods employ the fully nonlinear free surface boundary condition.However, none of the methods can handle overturning (breaking) waves. None of thefully nonlinear methods have included radiation effects in case C. Details aboutvarious formulations can be found in the literature given in the list of references.

It is difficult to make exact statements about the computational effort for thevarious methods employed as this among other, is related to the actual singularitydistribution applied, as well as the computer used. However, the following generaltrend is observed: the semi-analytical, linear method is by far the fastest method. Ituses only about 20 s on a state-of-the-art PC for case C. The frequency domainBEM, linear and second-order methods use a few minutes of CPU time per wave forthe cases A and B, while the nonlinear time domain methods use in the order onehour per wave period. Table 8 gives an overview of the actual combinations ofmethods and cases that have been used in the comparison.

4. Case A, circular column, monochromatic waves

4.1. Results from model tests

The model tests were performed in the towing tank of Marintek in Trondheim,[11] and [31]. The tank is 10 m wide and 10 m deep. The measurements were

Table 8

Cases considered by the different methods. (X means all mono- and bichromatic cases)

Case A Case B Case C

A M1, M2, M3 X

B X X

C X X X

D X X X

E M1, M2, M3 M1, M2, M3

F M1, M2 M1

G M1

H X

I M1a

a Computation failed.


performed in a model of scale 1:48.93. Sample pictures from the tests are included inFig. 3.

The actual wave heights measured during the model tests differed a little from thevalues specified in Table 2. The actual measured undisturbed mean doubleamplitudes are 3.94, 7.82 and 13.15 m, respectively. The corresponding maximumdouble amplitudes during the time interval used in the analysis of the model tests are4.07, 8.09 and 14.01 m, i.e. the maximum double amplitudes are all within 7% of themean double amplitudes. The length of the time records used in the analysiscorresponds to approximately 22 wave periods. Time traces of the undisturbed waveelevation are shown in Fig. 4.

Fig. 3. Front (left) and side view (right) of run-up on the circular column during model testing. Note the

vertical strings used for measurement of wave elevation (example from tests in a wave spectrum).

550 600 650 700 750-2

0

2

550 600 650 700 750-4

-2

0

2

4

550 600 650 700 750

-5

0

5

Time (sec)

Wa

ve

ele

vatio

n (

m)

Fig. 4. Undisturbed incident waves M1, M2 and M3 in the actual time interval of analysis.


In the following, the direction of the rows of wave gauges will be denoted byy0 ¼ y� 1801; where y is the angle given in Tables 1 and 3. This is to obtain angles inthe range 0–901. For some rows of wave gauges there are redundant measurement.The wave elevation is measured in 7451 and 722.51. This redundancy has been usedto check the quality of the measurements and to exclude records that proved to beunreliable. In the measurements presented, the mean crest elevation is normalized by0.5�H: H is the mean undisturbed wave height during the time window considered.A measure of the confidence limits of the measured response is given by the ratiobetween the maximum crest elevation measured and the mean crest elevation.Samples of the results are given in Table 9. For the lowest monochromatic wave(M1), only small variations between the maximum and mean crest elevations areobserved, while in the steepest wave (M3) differences up to 33% are observed. In thiscase wave breaking occurred. The differences between the specified and measuredwave heights are at most 6.7%.

4.2. Computed results

The computed crest elevations in regular waves are presented in Figs. 5–8. Theresults are scaled by H=2: H being the specified incident wave height. Theexperimental results are included in the figures.

For the lowest wave height not large differences between the different estimatesare obtained. Note, however, that an error has occurred for the measurementpresented in Fig. 7. The measured results in this case should be ignored. As the wavesteepness increases, the scatter in the predictions increases. The linear predictions(methods C and D) show very little radial variation in the run-up. Some of thenonlinear predictions (E and F) seem to be unstable or not converged for the twosteepest waves. In the steepest wave (M3) the estimates on the maximum run-up infront of the column varied between 1.27 and 1.8 times the incident wave amplitudewhile the measured value is approximately 1.6.

Considering Fig. 5, the 901 case, we observe that the wave elevation decreases aswe approach the wall of the cylinder. This effect can be explained by the fact that the

Table 9

Ratio between maximum and mean crest elevation in regular waves (M1, M2 and M3). Circular column.

Values taken in the time interval shown in Fig. 4. The locations A4i corresponds to measurements in front

of the column, while A1i corresponds to measurement 901 to the wave direction of propagation, see Fig. 1

Location Wave M1 Wave M2 Wave M3

A11 1.017 1.041 1.144

A12 1.018 1.040 1.140

A13 1.024 1.022 1.109

A14 1.017 1.026 1.124

A41 1.014 1.088 1.334

A42 1.013 1.084 1.153

A43 1.013 1.062 1.085

A44 1.015 1.036 1.211


Fig. 5. Maximum wave elevation along row A1 (901). Case A, monochromatic wave M1 (top), M2

(middle) and M3 (bottom).


Fig. 6. Maximum wave elevation along row A2 (451). Case A, monochromatic wave M1 (top), M2



Fig. 7. Maximum wave elevation along row A3 (22.51). Case A, monochromatic wave M1 (top), M2



Fig. 8. Maximum wave elevation along row A4 (01). Case A, monochromatic wave M1 (top), M2 (middle)

and M3 (bottom).


incident wave is fairly long compared to the diameter of the cylinder ðl=D ¼ 7:9Þ: Asa wave crest passes the cylinder, the horizontal particle velocity close to the cylinderwill be approximately twice the horizontal particle velocity far away from thecylinder. A rough estimate on the difference in wave elevation close to the cylinderand far away from the cylinder, Dz; due to the pressure drop resulting from thedifferences in velocities is obtained as

Dz

H=2E

3p2

T2

H

g: ð2Þ

Here T is the wave period. From this approximation we obtain Dz=ðH=2Þ ¼0:16; 0:29 and 0.47 for case M1–M3, respectively. From Fig. 5 we observe that thisprovides a reasonable estimate on the radial variation of the wave elevation. Thenonlinear computations show similar radial variations. As expected this variation isnot caught by linear computations (D and E).

For the 01 case, Fig. 8, we observe increased wave elevation as we approach thewall. In the limit of very short waves a linear amplification factor of two is expected.With the present wavelength, a linear amplification of 1.27 is obtained. Thenonlinear codes give considerable higher elevations and resemble the measurements.However, considerable scatter is observed between the computed results. Even ifconsiderable nonlinear amplification of the wave crest elevation is observed, none ofthe cases exhibits extreme run-up effects. In general, the scatter in the results is mostpronounced close to the cylinder wall.

5. Case B, square column, monochromatic waves

5.1. Results from model test

The model tests were performed in the same way as for case A. The incident wavetime histories as well as the time window of analysis are identical to case A. InTable 10 the ratio of the measured maximum crest elevation divided by mean crest

Table 10

Ratio between maximum and mean crest elevation in regular waves (M1, M2 and M3). Square column.

Values taken in the time interval shown in Fig. 4. The positions B3i corresponds to measurements in front

of the column, while B1i corresponds to measurement 901 to the wave direction of propagation, see Fig. 1

Location Wave M1 Wave M2 Wave M3

B11 1.030 1.035 1.092

B12 1.029 1.040 1.115

B13 1.037 1.053 1.058

B14 1.040 1.068 1.059

B31 1.032 1.046 1.131

B32 1.025 1.036 1.094

B33 1.024 1.025 1.038

B34 1.020 1.031 1.037


elevation for some locations are shown. For the lowest wave (M1) small variationsbetween the maximum and mean crest elevation is observed, while in the steepestwave differences up to 13% are obtained.


The computed crest elevations in regular waves are presented in Figs. 9–11. Forthe lowest wave height the scatter between the computed results is larger in case Bthan for case A. For 901 the two second-order methods (B and E) predictsignificantly higher run-up than the linear methods (C and D). For moderate wavesteepness, the second-order methods predict the measured crest elevations withreasonable accuracy, while for the steepest wave they overestimate the measuredvalues significantly. It should be investigated if flow separation have influenced theexperimental results in this case. For 451 and 01 (Figs. 10, and 11) the two second-order methods predict very different crest elevation. For 451 (E) is close to themeasured values, while for 01 (B) is closest. The linear predictions (C and D) showvery little radial variation in the crest elevation. This is similar to the circularcolumn. It should be noted that the linear predictions differ somewhat. This is mostlikely due to sensitivity to discretization of the body boundary. It is likely that a finerdiscretization together with more panels close to the free surface, would improve theresults. In the steepest wave (M3), estimates on the maximum run-up in front of thesquare column varied between 1.3 and 2.1 times the incident wave amplitude, whilethe measured value is 2.5. In the steepest wave, significant nonlinear run-up isobserved both for 01 and 451.

6. Case A, circular column, bichromatic waves

The bichromatic waves cases are specified in Table 2. The bichromatic cases werespecified to challenge the effect of wave–wave interaction in the nonlinearrepresentation of free surface elevation.


The input time series used in the analysis are shown in Figs. 12 and 13. In thesefigures, the total incident wave elevation as well as high pass and low pass filteredcomponents are shown. The irregular behavior of the high pass filtered componentindicates the presence of high frequency, nonlinear incident wave components. InFig. 14, a sample of the wave elevation along row A4 is shown. The maximumelevation is increasing towards the wall of the column. Further a time delay in themaximum elevation due to the horizontal position of the wave gauges is observed. Inanalyzing the experimental results for bichromatic waves, we used the maxima in thetime window considered. The results are normalized by the standard deviation of themeasured wave elevation.


Fig. 9. Maximum wave elevation along row B1 (901). Case B, monochromatic wave M1 (top), M2



550 600 650 700 750-10-6

-22

610

550 600 650 700 750-10

-6-2

26

10

550 600 650 700 750-10

-6-226

10

Time (sec)

wav

e el

evat

ion

(m)

Fig. 12. Incident wave record of bichromatic wave B2. Top: total wave, middle: low pass filtered, bottom:

high pass filtered time history (filter frequency 0.091 Hz).

550 600 650 700 750-16

-8

0

8

16

550 600 650 700 750-16

-8

0

8

16

550 600 650 700 750-16

-8

0

8

16

wav

e el

evat

ion

(m)

Time (sec)

Fig. 13. Incident wave record of bichromatic wave B3. Top: total wave, middle: low pass filtered, bottom:

high pass filtered time history (filter frequency 0.0833 Hz).


660 665 670 675 680 685 690 695 700

-10

-5

0

5

10

15

20A4.1 A4.2 A4.3 A4.4 Undisturbed

672 674 676

6

8

10

12

14

16

Time (sec)

Ele

vatio

n (m

)

Fig. 14. Measured crest elevation along row A4 (01) in bichromatic wave B2. Thick solid line: close to

column, thin solid line: about one radius in front of column. Upper right: details the local extreme.

Undisturbed incident wave is measured at the center of the cylinder.

660 665 670 675 680 685 690 695 700-15

-10

-5

0

5

10

15

20B3.2 B3.3 B3.4 Undisturbed

670 672 674

6

8

10

12

14

16B3.1

Time (sec)

Time (sec)

Ele

vatio

n (m

)

Ele

vatio

n (m

)

Fig. 15. Measured crest elevation along row B3 (01) in bichromatic wave B2. Thick solid line: close to

column, thin solid line: about one radius in front of the column. Upper right: details the local extreme.

Undisturbed incident wave is measured at the center of the cylinder.



The computed results for the circular column are shown in Figs. 16–19. For the22.51 case (Fig. 18) experimental results are not included for wave B1 and B3. Thesemeasurements were not reliable. The computed results are normalized by themeasured standard deviations of the measured incident wave elevations. These timerecords were provided to the participants as part of the input. The standarddeviation of the measured, incident wave is somewhat lower than what is obtainedby summation of two sinusoidal signals. This is illustrated in Table 11. Maximum10% difference between the two approaches is expected.

Only three sets of computations are available for the bichromatic waves. Of thesetwo are based upon linear computations. In general the second-order computation(B) seem to correspond better with the measurements than the linear estimates. Ingeneral the linear codes underestimate the maximum crest elevation. The second-order code overestimates the measured maximum crest elevation in front of thecylinder. At the side of the cylinder (901) the second-order code does underestimatethe radial variation of the maximum crest elevation. For the highest wave (B3) theestimates on the maximum run-up in front of the column varies from 2 to 3.5 timesthe standard deviation, while the measured value is 3.1. Both measurements and thesecond-order code identify significant nonlinear run-up in front of the cylinder.

7. Case B, square column, bichromatic waves


The model tests are performed in identical wave as for the circular column. InFig. 15 measured crest elevation along row B3 (01) in wave B2 is shown. In general,the measured maximum crest elevations for the square column are not very differentfrom those obtained for the circular cylinder. As single extreme values are compared,greater uncertainties in the data exist as compared to the regular wave case, whereaverage crest elevations were used.


The computed results for the square column in bichromatic waves are shown inFigs. 20–22. As for the circular column, only three sets of computed results areavailable. Two of the results (C and D) are based upon linear computations. Again,the results from the second-order method (B) in general have a similar trend as themeasurements. However, the magnitudes of the maximum crest elevation estimateshave large scatter. Except for 901, the linear estimates significantly underestimate therun-up. For the highest wave (B3) the estimates on the maximum run-up in front ofthe column varies from 2.0 to 3.7 times the standard deviation, while the measuredvalue is 3.2.


Fig. 16. Maximum wave elevation along row A1 (901). Case A, bichromatic wave B1 (top), B2 (middle)

and B3 (bottom).


Fig. 17. Maximum wave elevation along row A2 (451). Case A, bichromatic wave B1 (top), B2 (middle)

and B3 (bottom).


Fig. 18. Maximum wave elevation along row A3 (22.51). Case A, bichromatic wave B1 (top), B2 (middle)

and B3 (bottom).


Fig. 19. Maximum wave elevation along row A4 (01). Case A, bichromatic wave B1 (top), B2 (middle) and

B3 (bottom).


8. Case C, complete platform, monochromatic waves

The complete platform case, case C is described above. Model tests have beenperformed in scale 1/65. The tests were performed in the Ocean Basin Laboratory atMarintek with length�width=80 m� 50 m. A soft mooring system with horizontalrestoring force as given above was used in the tests. The locations 1–6 of the airgapmeasurements are specified in Table 6. The wave gauges were platform mounted.The measurements obtained are thus the instantaneous distances from deck level tothe free surface, including the effect of platform motion. To transform thesemeasurements to relative wave elevation, the following conversion has beenperformed:

ZðtÞ ¼ AG0 � AGðtÞ: ð3Þ

Here AG0 is the airgap in calm water, while AGðtÞ is the measured instantaneousairgap. By this definition the relative wave elevation includes the effect of platformmotion and is measured relative to the incident calm water level. For the wave periodconsidered, the computed vertical motion of the platform at the various locations forthe wave gauges range from 0.13 m/m to 0.27 m/m incident wave amplitude. Thelowest value is valid for the front locations, while the highest value is valid for therear location. These values indicate the importance of ignoring the platform motionsin the calculations.


Time histories of the relative wave elevations are shown in Figs. 23–26. For thetwo highest wave heights, significant nonlinearities are observed in the records. Forthe highest wave (Fig. 26) wave breaking occurred. In this case the wave hits the deck(elevation 25 m above calm free surface) in point 4. Fourier analyses of the measuredrelative wave elevation reveals that, in addition to first and second harmonics,several of the records contain significant third harmonics. This indicates that second-order theory may be insufficient to describe the wave elevation.


In Figs. 27 and 28 the computed results are presented together with themeasurements. The measurements are maximum relative crest elevations averaged

Table 11

Measured standard deviation of incident bichromatic wave, sm compared to the standard deviation, sx of

a process given from x ¼ H1=2 cosð2pt=T1Þ þ H2=2 cos 2ðpt=T2Þ

B1 B2 B3

sm 1.688 5.095 8.063

sx 1.740 5.685 8.257


Fig. 20. Maximum wave elevation along row B1 (901). Case B, bichromatic wave B1 (top), B2 (middle)

and B3 (bottom).


Fig. 21. Maximum wave elevation along row B2 (451). Case B, bichromatic wave B1 (top), B2 (middle)

and B3 (bottom).


Fig. 22. Maximum wave elevation along row B3 (01). Case B, bichromatic wave B1 (top), B2 (middle) and

B3 (bottom).


Fig. 23. Case C. Sections of the time series of the wave elevation under the platform in points 1–6.

Incident wave height 6.09 m.




Fig. 27. Case C. Relative wave elevation in points 1–3. Scaled by H=2:


Fig. 28. Case C. Relative wave elevation in points 1–3. Scaled by H=2:


over 16 wave periods. The results are normalized by z0 ¼ H=2: Method (A)is a second-order frequency domain method while the remaining three methodsused in this case are all linear methods. In the estimates from method (A),platform motions are not included. We observe that the estimates based uponthe second-order method are not necessarily better than the linear estimates. Thismay, at least partly be related to the treatment of the platform motion. However,large scatter is also observed in the linear estimates. Some of the airgaplocations have a close proximity to the platform columns. An unfavorablediscretization of the platform geometry may thus affect the results. This statementis valid for method (C) and (D). Method (H) is semi-analytic. Thus the proxi-mity effect should not matter in this case. From Figs. 23–26, it is observed thatthe measured relative wave elevation is very asymmetric. This asymmetry isnot accounted for in the linear methods. From experience, the platform motionitself is fairly symmetric. Thus most of the asymmetry is related to the waveelevation.

Point 1 is the center point under the platform. A significant effect of higherharmonics is observed, Figs. 25 and 26. Also the nonlinear computations reveallarge nonlinearities, Fig. 27, top. However, the nonlinear effect does not signi-ficantly effect the amplitude of the wave elevation. In point 2 (midway betweenthe rear columns) also large nonlinearities are observed in the measurements.This is most likely due to local resonant effect between the columns. A wavewith period half the period of the incident wave has a wavelength approxi-mately equal to the distance between the columns. In spite of the largenonlinear effects in the measurements, the linear predictions are closer to themeasurements than the results from the nonlinear method. In point 3 (midwaybetween front and rear column) no large second-order components areobserved in the measurements. The linear and nonlinear computations alsogive fairly similar results. In points 4 and 5 (both located close to and in frontof rear column), the asymmetry in the measurements is large. This seems to bedue to large second-order components in the wave elevations that at thesepoints add to the amplitudes of the waves. This effect is caught neither bythe linear nor by the nonlinear computations. The effect may be explainedby a combination of resonance effects between the columns, as discussed above,and a local run-up effect along the column. (The wave gauges are mountedonly 3 m=0.1 diameter from the wall of the column). In point 6 (in front ofthe front column) both, one of the linear methods as well as the nonlinearmethods seem to estimate the run-up fairly accurately. However, for point 5as well as for point 6, two of the linear predictions are almost a factor twoless than the other predictions. The reasons for these erroneous results arenot found. For method (D) a possible explanation could be that thediscretization of the body combined with the close proximity cause errors in theresults. This does, however, not explain the results for the semi-analytic method (H).From McCamy and Fuchs theory a linear amplification at location 6 should bebetween 1.3 and 1.4. Phasing of the combined heave–pitch platform motion willaffect the results.


9. Summary and conclusions

In the comparative study computation of airgap and run-up six universities andcompanies have contributed. Altogether nine different computer codes have beenemployed. Three different cases have been considered: two rigid columns, one withcircular cross-section, the other with a square cross-section with rounded corners.The third case was a complete semi-submersible with four columns.

The codes used are all based upon potential theory. They include linear frequencydomain methods, second-order time and frequency domain methods as well as ‘‘fullynonlinear’’ time domain methods. Not all the codes have been employed in all cases.For all cases considered, experimental results are available for comparison. Themeasurements are considered to be reliable, as several measurements are redundant.

Computation of wave crest elevation close to a body is an extremely difficult task.From a practical point of view we are interested in the most extreme events. Theseare the most difficult cases to handle. As could be expected, the comparisons exhibita significant scatter among the computed results.

For the circular column in monochromatic waves, it is observed that the linearmethods fail to predict reasonable crest elevations except for at very low wavesteepness. The nonlinear methods predict trends in wave crest elevation similar to themeasurements both with respect to wave steepness and proximity to the wall.However, some scatter is observed between the methods. For the ‘‘fully nonlinear’’methods, the results in some cases do not seem to be converged. One ‘‘fullynonlinear’’ method failed to produce any results.

For the square column the scatter among the results are larger than for the circularcolumn. In most cases the nonlinear methods provide a correct trend in the resultsand better predictions than the linear methods. Both at the front face of the squarecolumn as well as at the corner there are significant challenges in computing the crestelevation. Significant nonlinear run-up effects are observed at these positions. At thecorner, a ‘‘fully nonlinear’’ method provide results close to the measured values,while at the front face a second-order method provides the best results.

In bichromatic waves, the trend is similar as in monochromatic waves: the second-order method employed gives in general a correct trend in the results, but not alwaysbetter numerical estimate than the linear methods.

For the complete platform significant resonance effects between the columns affectthe airgap. The resonance effect was not present in the single column case. For lowincident wave heights, both linear and nonlinear methods predict the airgap withinapproximately 30% of the measured value. As the steepness of the incident waveincreases, all methods fail to predict the airgap/run-up at the points of major interest.The nonlinear computations do not always provide better results than the linearcomputations.

The reasons for the scatter in the results are not fully understood. Butcontributions to the scatter may come from numerical discretization, ignoringplatform motion as well as nonlinearities of higher order than second order.

It is recommended that more effort is put into understanding details of the run-upphenomenon. This can be done by carefully observing the details of the flow as well


as by careful numerical modeling. The nonlinear resonance between columns shouldbe studied in more detail.

Acknowledgements

The contributors to this study are: BMT Fluid Mechanics Limited, Boston MarineConsulting, Laboratoire de Mecanique Des Fluides Ecole Centrale Nantes, NobleDenton Europe, Ltd., Department of Ocean Engineering, MIT, Twente Instituteof Mechanics, Univ. of Twente. They are all acknowledged for their importantcontributions to this study. They performed the work without any compensation.Marintek and in particular Carl Trygve Stansberg is acknowledged for his importantcontribution in designing the experiments and subsequent interpretation of theresults. Norsk Hydro Research Center is acknowledged for allocating time making itpossible to complete this study.

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