Kinematics of extreme waves in deep water

Transcription

1 Alied Ocean Research 25 (2003) Kinematics of extreme waves in dee water John Grue*, Didier Clamond, Morten Huseby, Atle Jensen Mechanics Division, Deartment of Mathematics, Univeristy of Oslo, P.O. Box 1053, Blindern, Oslo 0316, Norway Received 1 January 2003; revised 2 February 2004; acceted 1 March 2004 Available online 20 June 2004 Abstract The velocity rofiles under crest of a total of 62 different stee wave events in dee water are measured in laboratory using article image velocimetry. The waves take lace in the leading unsteady art of a wave train, focusing wave fields and random wave series. Comlementary fully nonlinear theoretical/numerical wave comutations are erformed. The exerimental velocities have been ut on a nondimensional form in the following way: from the wave record (at a fixed oint) the (local) trough-to-trough eriod, T TT and the maximal elevation above mean water level, h m of an individual stee wave event are identified. The local wavenumber, k and an estimate of the wave sloe, e are evaluated from v 2 =ðgkþ ¼1 þ e 2 ; kh m ¼ e þ 1 2 e2 þ 1 2 e3 ; where v ¼ 2=T TT and g denotes the acceleration of gravity. A reference fluid velocity, e ffiffiffiffi g=k is then defined. Dee water waves with a fluid velocity u to 75% of the estimated wave seed are measured. The corresonding kh m is A strong collase of the nondimensional exerimental velocity rofiles is found. This is also true with the fully nonlinear comutations of transient waves. There is excellent agreement between the resent measurements and reviously ublished Laser Doler Anemometry data. A surrising result, obtained by comarison, is that the nondimensional exerimental velocities fit with the exonential rofile, i.e. e ky ; y the vertical coordinate, with y ¼ 0 in the mean water level. q 2004 Elsevier Ltd. All rights reserved. Keywords: Wave kinematics; Extreme waves; PIV 1. Introduction Enhanced evidence and descrition of the kinematics during stee wave events at sea are requested by the offshore and ocean engineering industry. The velocities in stee waves are required for subsequent analysis of loads on, e.g. shis, offshore latforms, tension legs and risers. Desite the numerous studies on the subject, roer knowledge of kinematics of stee irregular ocean waves is still lacking. This rovides the motivation of the resent investigation. We comare exerimental velocity fields due to sets of random wave trains, stee wave events due to focusing waves, stee wave events due to the unsteady leading art of a eriodic wave train, and the velocities in comutations of stee transient waves. We comare recise Particle Image Velocimetry (PIV) measurements in laboratory and a fully nonlinear modeling. The velocity immediately below the wave crest is focused. While irrotational flow theory may redict the wave kinematics u to breaking, the theory has shortcomings * Corresonding author. Tel.: þ ; fax: þ address: johng@math.uio.no (J. Grue). beyond this limit. This is where exeriments become articularly valuable since they are not limited by wave breaking. For examle, a series of breaking wave events may take lace during the long irregular wave tests undertaken here, recording several strong wave events Previous exerimental works We begin with a short summary of revious exerimental work. Large scale observations of the kinematics of storm waves are given by, e.g. Buckley and Stavovy [1] and Forristall [2]. LDV (Laser Doler Velocimetry) laboratory exeriments are carried out, most notably by Skjelbreia et al. [3,4], Kim et al. [5], Longridge et al. [6] and Baldock et al. [7]. These works include descritions of theoretical models. In the large scale FULWACK exeriment [2], the main urose was to obtain velocity measurements at locations relatively high-u in the waves. They used current meters located at 26, 16, 6 ft (8.2, 5, 1.9 m) above mean sea level. The largest observed seed at the to current meter was ft/s (6.5 m/s). Kim et al. [5] measured the kinematics due to focusing waves in a laboratory wave tank with moderately dee water /$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi: /j.aor

2 356 J. Grue et al. / Alied Ocean Research 25 (2003) using LDA (Laser Doler Anemometry). Recordings were made at three levels between the mean water line and the crest. Their most extreme wave had an estimated wave sloe of Very large horizontal velocity was documented at the crest of the wave, u to 64% of the measured hase velocity. (The effect of a finite water deth is rather ronounced in the exeriments.) They also estimated the acceleration field including the convective acceleration using finite difference technique. Comarisons were made with stretching methods, which they found to greatly underestimate the fluid velocity at the crest of the transient wave. LDV measurements of stee wave events of an irregular sea, at several vertical ositions of the wave field, were erformed by Skjelbreia et al. [3,4]. The local wave sloe was u to about Major emhasis was given to the surface zone. They measured the fluid velocity at levels close to the crest, half way and one-quarter way between the mean water level and crest, concluding that the stretching method of Wheeler [8] comared better with the measurements of the horizontal velocity below wave crests than Stokes fifth-order model. No accelerations were obtained. Their exeriments were related to small and moderate waves with relatively few measurements above mean water level. (Our results do not suort the Wheeler stretching method.) Laboratory measurements of wave kinematics of four irregular wave trains using LDA were described by Longridge et al. [6]. They comared the results with a hybrid wave model. The irregular wave fields were roduced using the Pierson-Moskowitz (PM) and the JONSWAP sectra. The local wave sloe in the wave events they considered was in the range The aer focused on the velocity and acceleration fields induced by relatively stee wave events. The fluid velocity was measured u to about half way between the mean water level and the crest. The measurements were comared with linear extraolation, Wheeler stretching and a hybrid wave model, finding that the linear extraolation was the suerior one for the waves with strongest nonlinearity. Accelerations were obtained below the mean water line. Baldock et al. [7] erformed rather detailed and recise sets of wave tank measurements of focusing waves. Both the surface elevation and the induced fluid velocities were resented. The fluid velocities were obtained using LDA. We estimate the wave sloe of their largest wave to be about 0.29 (using the rocedure given in Section 2.4). Baldock et al. noted that their measured velocities close to the wave crest significantly exceeded the revious velocity recordings by, e.g. Skjelbreia et al. [3,4]. They further noted that their measurements could not be exlained by stretching models, for examle. (We find here an excellent agreement between the resent PIV measurements of several sets of stee wave events and the LDA measurements documented by Baldock et al.) 1.2. Plunging breakers Strong lungers were observed in the exerimental works by Kim et al. [5] (kh ¼ 2:0; kh m ¼ 0:33; where k denotes wavenumber, h the water deth and h m the maximal elevation of the event) and Skyner [9] (kh ¼ 2:6; kh m ¼ 0:36). In the exeriments by Skyner the recorded fluid velocity was found to exceed the (nonlinear) wave seed, although the latter is not well defined for strongly nonlinear waves. We note that the velocity field beneath unsteady two-dimensional water waves have been accurately comuted u to the commencement of overturning by Dold and Peregrine [10] and Peregrine [11]. Recent exeriments on a silling breaker show that some water exceeds the wave seed, but that the mean velocity may well be smaller [12]. Although lunging breakers are not so common in dee water, these may occur. In the steeest wave event we have observed in the resent exeriments, the value of kh is 5. This wave is develoing into a weak breaker. The estimated local wavenumber times maximal elevation of the event is kh m ¼ 0:62 and the largest measured fluid velocity under crest is 75% of the hase velocity. Data for stee wave events are comared in Section Shortcomings of revious work and oints requested Theories of steady and random wave fields were reviewed by Gudmestad [13] concluding that regular waves are very accurately redicted by nonlinear theory. He noted that the kinematics in random seas, or in sea states where unsteady very stee waves occur, is less well redicted, however. Shortcomings of revious work and main oints requested include: there is lack of resolution in velocity field measurements above the mean water level, articularly in the to and near the ti of the wave. Denser velocity measurements are required to reduce the uncertainty in the acceleration estimates. More laboratory (and full scale) measurements are requested to comlement the relatively scarce selection of waves that are studied in revious works. An imortant oint is that most of the exeriments so far have been carried out for relatively small or moderately stee waves. Several of these measurements are found to comare well with redictions using the second-order model for irregular seas [14] or extensions of this model [15 17]. A good agreement between model and exeriment is not seen for very stee waves, however. The Wheeler stretching method, used e.g. in engineering ractice in the Gulf of Mexico, is concluded to give relevant redictions of the horizontal velocity below a wave crest. The ractice is based on the exerimental results by Skjelbreia et al. [3,4]. In contrast to this, Gudmestad [13] noted that velocities under stee random waves comare less well with the stretching models. Measurements then show larger velocities under the crest than redicted by

3 J. Grue et al. / Alied Ocean Research 25 (2003) models. According to Gudmestad, the kinematics of freak waves and waves which are near breaking, is not well redicted by any existing theory, and that further research is needed with resect to the kinematics of these waves. A model that takes into account the fully nonlinear wave wave interaction is requested Focus of the resent work Our focus is the kinematics of stee wave events which may occur on the surface of the ocean. We comare theoretical and numerical redictions using a recent fully nonlinear wave model (which has no limitations with regard to nonlinearity, disersion or resolution) with high resolution PIV exeriments in a laboratory wave tank. We consider several different wave scenarios in a total of 62 different stee wave events in the wave tank. The total number of PIV/wave exeriments reorted here is 122, counting all reetitions which are carried out. Our main objective is to identify the velocity rofile under crest of a stee wave event. The wave event is a result of a rocess on the surface of the fluid where the wave field, from an initial state, has develoed due to nonlinearity and disersion. We assume that this rocess has gone on during sufficiently long time such that the initial conditions are forgotten. We here focus on the kinematics of several stee wave events resulting from four different generation rocedures. Three of the rocedures are ursued exerimentally in the laboratory. A fourth one results from numerical simulations. Most of the exerimental waves can be considered to be dee water waves (kh larger than about 3). In a few of the exeriments the finite deth of the water introduces a small, systematic change in the wave rofiles comared to the dee water results, however. This tyically occurs for kh, 2 and is discussed in Section We comare our measurements with those in Refs. [5,7, 9]. We further discuss the large-scale observations of the FULWACK exeriment in view of the resent results, see Section 5. The waves studied here are significantly larger than the waves documented in Refs. [3,4,6], and comarisons with these works are not erformed. We aim at a general way to characterize the stee wave events in the exeriments. The urose is to interret and communicate the results. More recisely, from the wave record we identify a local wave eriod and a maximal excursion of an individual stee wave event. From these two arameters we rovide an estimate of the local wave sloe, the wavenumber and a characteristic fluid velocity of the event. The rocedure is documented in full in Section Theoretical models Third-order Stokes waves, fully nonlinear simulations of Stokes waves and a general fully nonlinear, unsteady wave model rovide references for the exeriments. The models are first discussed Third-order Stokes waves The theory of third-order Stokes waves is valid for eriodic rogressive irrotational wave motion with a moderate wave sloe in dee water. Let {x; y; t} be the horizontal, uward vertical and time variables, resectively, and let the mean water level be at y ¼ 0 and the free surface at y ¼ h: Let f denote the velocity otential such that 7f! 0asy! 21: Denoting {g; k; v} the acceleration due to gravity, the wavenumber and the angular frequency, a variant of Stokes third-order aroximation reads kf ffiffiffiffi ¼ e e ky sin u þ Oðe 4 Þ; ð1þ g=k kh ¼ 1 þ 1 8 e 2 e cos u þ 1 2 e2 cos 2u þ 3 8 e3 cos 3u þ Oðe 4 Þ; v 2 gk ¼ 1 þ e 2 þ Oðe 3 Þ; kh m ¼ e þ 1 2 e 2 þ 1 2 e 2 þ Oðe 4 Þ; where u ¼ kx 2 vt denotes the hase function and the latter equation relates e to the maximum surface elevation times the wavenumber. (In the exansion resented here the amlitude of the fundamental frequency of the fluid velocity is chosen as small arameter. The half of the wave height is determined by ½e þ 1 2 e 3 þ Oðe 4 ÞŠ=k:) Eq. (1) shows that an exact Stokes wave velocity field is well aroximated by linear theory, even for relatively stee waves. For this reason, linear wave theory is widely used to estimate the wave induced velocity field above and below the mean water level. Indeed, the aroximated velocity otential (1) suggests that the velocity field below the crest is almost self-similar (within a reasonable accuracy). It is temting to evaluate a dimensionless horizontal velocity below the wave crest of the form u=ðe ffiffiffiffi g=k Þ; where u denotes the horizontal velocity Fully nonlinear Stokes waves Exact (fully nonlinear) comutations of (steady) Stokes waves are included for illustrative uroses. The comutations are facilitated by Fenton s rogram [18] (roviding an accuracy of at least five digits for the results shown here). The comutations in Fig. 1a of the velocity rofile below crest are very close to the exonential rofile when the wave is moderately stee (e u to about 0.2). Such rofiles have also been exerimentally obtained in PIV exeriments for wave sloe u to 0.16 [19]. The theoretical results are far from uniform when e is in the range (Fig. 1a). ð2þ ð3þ ð4þ

4 358 J. Grue et al. / Alied Ocean Research 25 (2003) Fig. 1. Normalized velocity rofiles in fully nonlinear Stokes waves (comuted as described in Section 2.2) (a) and freak waves obtained (comuted as described in Sections 2.3) (b). Values of e in lot (a) (from largest to smallest): 0.360, 0.357, 0.351, 0.333, 0.298, 0.258, Values of e in lot (b): 0.290, 0.290, 0.285, 0.267, 0.236, 0.212, Dotted line: e ky : We shall see below that the nondimensional exerimental velocities, with kh m u to 0.62 (and e u to 0.46), agree better with estimates from Stokes third-order solution than with the fully nonlinear Stokes solution, for the stee waves. The trick is to identify a local k and e of the unsteady wave using the rocedure of Section Fully nonlinear simulations of unsteady waves In recent comutations, emloying a raid fully nonlinear transient method, very stee wave events have been obtained [20 22]. In brief, the theoretical/numerical scheme integrates the rognostic equations. These result from the kinematic and dynamic boundary conditions at the free surface, i.e. h t þ ~c x ¼ 0; ~f t þ gh þ 1 ~f 2 x 2 ~c 2 x þ 2h x ~f x ~c x 2 1 þ h 2 ¼ 0; x where the tildes denote the quantities evaluated at the free surface y ¼ h; and c denotes the stream function. The equation connecting the (scaled) normal velocity at the free surface, ~c x ; h and ~f x ; is obtained by solving the Lalace equation in the fluid domain. The resulting equation is on the form ~c x ¼ H{ ~f x } þ x {h ~f x } þ x {H{hH{ ~f x }}} ( þ H ð1 21 ð1 D 2 ðd 2 h x Þ ~f 0 x dx þ D 2 x 0 2 x ) DðD 2 h x Þ ~c 0 x dx 0 1 þ D 2 x 0 2 x; where H denotes Hilbert transform, D ¼ðh 0 2 hþ=ðx 0 2 xþ; ~f ¼ ~fðx; tþ; ~f 0 ¼ ~fðx 0 ; tþ; etc. The latter imlicit equation can be solved numerically very quickly via Fast Fourier Transform and truncated integrations. Numerical simulations of very stee (freak) waves are obtained as in Refs. [21,22]: the initial state of the wave field is secified into two stes. First an exact steady Stokes wave, with wavenumber k 0 and amlitude a 0 (a 0 half the total wave height) was comuted. Secondly, the surface elevation and the tangential velocity at the ffiffi surface were multilied by the bell function such ½e 0 2a0 k0ðx 2 2 x 0 ÞŠ; where the arameter e 0 determines the length of the acket. The case e 0 ¼ 1 corresonds to an exact soliton solution of the nonlinear Schrödinger equation. This initial condition, in the form of a localized wave acket, is inut to the fully nonlinear numerical scheme. With a 0 k 0 ¼ 0:091 and e 0 ¼ 0:263 a very stee wave event is roduced after 155 wave eriods. The maximal wave elevation h m times k 0 is then k 0 h m ¼ 0:29: Other relatively stee wave events (with smaller k 0 h m ) are roduced using the same rocedure, with less strong initial conditions. The comutational domain involves 128 wavelengths and the carrier wave is discretized over 32 nodes er wavelength. This means that all harmonics u to the 15th are resolved, and that 128 Fourier modes are included in the sectral band ½k k 0; k 0 þ 1 2 k 0Š: The wave induced Stokes drift close to the free surface is inherent in the fully nonlinear formulation. This is also true with regard to a return flow beneath the wave grou. The latter contributes to an almost vanishing time-averaged horizontal mass flux during the wave motion. This is indicated in the fully nonlinear simulations by the integral of the stream function along the free surface, i.e. Ð 1 ~ 21 c dx ¼ Ð 1 Ð h u dydx: This integral is an order of magnitude smaller than the estimated Stokes drift indicating that a return flow beneath the wave grou balances the Stokes drift, aroximately. The return flow is caused by the radiation stress [14] and is reroduced in second-order theories [23 25].

5 J. Grue et al. / Alied Ocean Research 25 (2003) Velocity rofile on nondimensional form We are seeking a nondimensional reresentation of the velocity rofile under crest of a stee unsteady wave event. It is temting to test out the following rocedure: 1. In a stee wave event we identify, from the time history of the surface elevation at a fixed geometrical location, the trough-to-trough eriod, T TT and the maximal elevation, h m of the event. We then define the local angular frequency by v ¼ 2=T TT : 2. We next comute the wavenumber k and a measure of the wave sloe e solving numerically the system of equations v 2 =ðgkþ ¼1 þ e 2 ; kh m ¼ e þ 1 2 e2 þ 1 2 e 3 ; ð5þ which result from a truncation of Eqs. (3) and (4). The half of the wave height is estimated by ½e þ 1 2 e3 Š=k: Such a rocedure makes it ossible to extract the local wave length from the time record of the wave elevation in one oint. Using the fully numerical simulations we have comared 2=k; obtained by this method, with the trough-to-trough wavelength, L TT obtained directly from the nonlinear wave comutations. Several stee (freak) wave events have been tested. We found a relative difference of about 3.5% for the larger waves, with estimated kh m ¼ 0:34: The agreement was better for the smaller waves. We then consider the horizontal velocity rofile below the simulated stee waves. Nondimensional velocity rofiles are obtained dividing the fluid velocity by u ref ¼ e ffiffiffiffi g=k using the rocedure outlined above. Results from the comutations with e in the range are resented in Fig. 1b. A collase of the velocity rofiles is observed. Comarison is also made with the exonential rofile. The nondimensional velocities resulting from the transient wave comutations, in the region above mean water line, are closer to the exonential rofile than observed in fully nonlinear comutations of Stokes waves, when e is large. This rocedure to obtain nondimensional velocity fields aears relevant to stee, unsteady wave events. It is simle and robust in use. The resulting nondimensional exerimental velocities are exected to be relatively close. We have also tested out corresonding equations including terms u to fifth order, finding only very minor corrections. The main focus here is the velocities above the mean water line, with main emhasis on the flow close to the wave crest. In what follows we use as theoretical reference the largest freak wave comutation with e ¼ 0:29 and kh m ¼ 0:34: In addition we comare with the exonential rofile. Different rocedures to obtain a local wave eriod have been tested: zero ucrossing, zero downcrossing, etc. It aears, after some tests, that the trough-to-trough eriod gives the more consistent results. The characterization of individual waves of the wave field by the local wavenumber and wave eriod can mathematically be justified when the wave sectrum is sufficiently narrowbanded. Such a rocedure reresents a first ste also in the more general (and broadbanded) case. We have in our laboratory measured the nonlinear disersion relation in focusing wave grous finding that the measurements of wave eriod and wave length fit with nonlinear theory of slowly varying wave trains [26,27]. 3. Exerimental environment and rocedure The exeriments were carried out in a wave tank 24.6 m long and 0.5 m wide in the Hydrodynamics Laboratory at the University of Oslo. It was filled with water to a deth of 0.6 m in the exeriments with the leading wave of a wave grou (Section 4.1), and to a deth of 0.72 m in the focusing wave and random wave exeriments (Sections 4.2 and 4.3). In one end of the tank there is a hydraulic iston wavemaker with movements controlled by a comuter. The udate rate of the wave maker and the samling rate of the data acquisition is 1000 Hz. At the oosite end of the wave tank there is an absorbing beach which reflects less than 3% of the amlitude of the incoming waves. The kinematics of the waves is obtained by emloying an extended PIV system [19]. The system, articularly designed to measure accelerations, consists of two CCD cameras, a scanning laser beam and a synchronizer controlled and monitored by a comuter. The laser source is a CW argon ion laser, sufficiently owerful (10 W) to rovide light for sequences of recorded image airs. The two high-sensitive cooled PCO Sensicam cameras have a resolution of ixels with 12-bit digital outut. A high-seed acousto-otic modulator is used to shut off the CW laser after two scans of the flow have been catured by the first camera. The modulator switches the beam back on for the second velocity measurement after a rogrammed delay. Synchronization of the cameras and beam modulator with the scanning beam system is achieved with a urosebuilt multi-channel synchronizer device and oerated from an integrated modular tree based acquisition and rocessing software system. We resent here velocity measurements from the first camera. First, the wave elevation of the actual wave field was recorded at secified locations using wave gauges. Individual stee wave events were then identified from the wave records. Each exeriment was reeated measuring the velocity field under the large wave(s) using PIV. The system was triggered at the time instant when the wave crest was in the middle of the interrogation window. Several reetitions with the same wave conditions were erformed. The water in the tank was seeded with olyamid articles with diameter aroximately 50 mm. The field of view was

6 360 J. Grue et al. / Alied Ocean Research 25 (2003) cm 20 cm in all the runs. In most exeriments an interrogation window of ixels with an overla of 50% was used. In some exeriments the interrogation window was ixels. More details of the PIV system and the data rocessing rocedures may be found in Ref. [19]. We obtain from the velocimetry the velocity field of the waves in vicinity of the crest. The wave events are characterized by the local eriod and the local maximal elevation, both obtained from the wave record as described in Section 2.4. The arameter e; wavenumber and characteristic velocity are obtained from Eq. (5). Nondimensional velocities are then evaluated. Physical velocities are obtained from the resented results multilying by e ffiffiffiffi g=k : 4. Exerimental results 4.1. Leading wave of a wave train A sinusoidal motion of the wave maker with frequency v 0 generates a wave train which has a transient leading art followed by eriodic waves. (The motion of the wave maker has in these exeriments a constant amlitude, aart from a short initial eriod of 1 s when an amlitude function rises from 0 to the value of The amlitude function has the form of a tanh-function.) We ay here attention to the transient leading art of the wave fields. Relatively strong unsteady wave events take lace there. An examle is visualized in Fig. 2a. We investigate the kinematics of all of the largest wave events that are recorded in these series of Fig. 2. Wave fields. (a) Leading wave of a wave grou. (b) Focusing wave. (c) Sectrum of random waves obtained from wave record vs. inut sectrum (smooth line). v 2 H s =2g ¼ 0:10; T ¼ 1:04 s, (T 2 ¼ 0:777T ).

7 J. Grue et al. / Alied Ocean Research 25 (2003) exeriments, counting u to 13 in total. First the wave elevation at a secified location was recorded using a wave gauge. Next the run was reeated measuring the velocity field under the wave using PIV. In this set of exeriments, the arameter e in the different events is in the range , the inverse of the wave eriod in the range s 21, and the wavenumber times the water deth, kh in the range 2 5 (h ¼ 0:6 m). The nondimensional velocity rofiles of the 13 different waves are lotted together in Fig. 3a. A surrising collase of the data is observed. There is only very small scatter in the results. The exerimental rofiles, ut on nondimensional form, are seen to follow the velocity rofile of the theoretical freak wave comuted in Sections 2.3 and 2.4 (Fig. 1b and c). It is also good fit between the exeriments and the exonential rofile Focusing waves Fig. 3. Horizontal velocity rofiles below wave crest. (a) Thirteen large wave events in the leading unsteady art of a wave grou, with 0:213, e, 0:348: (b) Nineteen large wave events of focusing wave grous, with 0:217, e, 0:463: (c) Largest focusing wave event in: resent measurements, e ¼ 0:46; kh m ¼ 0:6 (S), LDA measurements by Baldock et al. [7] (þ), second-order model redictions reroduced from Ref. [7] ðwþ: Solid line: Velocity rofile in freak wave comutation with e ¼ 0:29: Dotted line: e ky : Dashed line: Wheeler method for wave with e ¼ 0:46; kh m ¼ 0:6: Several sets of focusing waves have been generated in the wave tank. In these runs the water deth was 0.72 m. The motion of the wave maker was given by jðtþ ¼aðtÞ sin½vðtþtš; where vðtþ=2 was decreased linearly from 1.6 to 0.96 s 21. The amlitude function had the form of aðtþ ¼ ^a 0 ð2 8 =3 3 Þ ½ðt=t s Þ 3 2 ðt=t s Þ 4 Š; with t=t s, 1; where t s was in the range s. (^a 0 denotes a constant amlitude which could be varied from exeriment to exeriment.) The large waves were recorded at a time instant the interval s after the wave maker motion was started. An examle of a stee wave is visualized in Fig. 2b. A total of 19 different focusing wave events were investigated. The arameter e (see Eq. (5)) in the different events is in the range , the inverse of the wave eriod in the range s 21, and the wavenumber times the water deth kh in the range (h ¼ 0:72 m). (The longest waves in these runs are not dee water waves. The steeest waves, that are the most interesting ones, are dee water waves, however.) The nondimensional velocity rofiles of the 19 different wave events are lotted together in Fig. 3b. There is a strong collase of the data. The exerimental rofiles follow the velocity rofile of the theoretical freak wave comuted in Sections 2.3 and 2.4 (Fig. 1b and c). Further, a good comarison between the nondimensional velocities and e ky is seen. There is only very small scatter in the exerimental results. We continue in Section 5 below a discussion of the very large velocities recorded in the focusing wave events. We find that the fluid velocity is u to about 75% of the estimated hase seed of the wave, with corresonding kh m ¼ 0:62 (and kh ¼ 5). Some surious velocity vectors aear in some of the measurements close to the free surface. These measurements indicate occurrence of breaking in the to of the wave. The breaking rocesses observed in the exeriments will not be discussed further here. We note that the velocity estimates in the thin breaking zones are uncertain Random waves Stee wave events in random seas using the JONSWAP sectrum are then measured. The sectrum is characterized by the significant wave height, H s ; the eak eriod, T ; and the eak enhancement factor, g ¼ 3:3: A total of six

8 362 J. Grue et al. / Alied Ocean Research 25 (2003) different time series were roduced, each with a length of 10 min. The inut sectrum of the wave field comare excellent with the sectrum roduced from the recorded elevation in the tank (Fig. 2c). An effect of a small reflection from the beach is not visible in the figure. The velocity fields in a total of five stee wave events in each of the six series have been analysed using PIV. The ranges of the arameter, e; inverse eriod, f ; and wavenumber of the individual waves times the water deth, kh; are indicated in the table below (where v ¼ 2 =T ; h ¼ 0:72 m). H s (cm) T (s) v 2 H s =2g e f (s 21 ) kh Series Series Series Series Series Series Velocity rofiles Each of the series in the random wave exeriments had three reetitions, all with the same triggering of the PIV system. The exeriments were very reeatable. The velocity data from the reeated runs exhibit a sread of about lus/minus one grid oint (there are 40 grid oints in each direction), excet in some of the breaking wave cases. The results from series 1 show an all over collase of the velocity rofiles when they are ut on nondimensional form. Moreover, the measurements are very close to the fully nonlinear comutations with e ¼ 0:29 and the exonential rofile, see Fig. 4. The corresondence is true for wave Fig. 4. Horizontal velocity rofiles below wave crest. Five wave events of series 1 ðsþ: Solid line: Velocity rofile in freak wave comutation with e ¼ 0:29: Dotted line: e ky : Recording times: , , , , s. Fig. 5. Same as Fig. 4, but series 4. Recording times: , , , , s. events that both are early and late in the series and is articularly good in the to of the wave. The events in series 1 are relatively stee, and the water is dee (0:19, e, 0:34; 3:9, kh, 4:6). The wave data in series 4 are in many ways comarable to those in series 1. The nondimensional water deth is in the same range in the two series. The wave sloe is somewhat higher in the events in series 4 ð0:26, e, 0:4Þ than in series 1. There is again a collase of the nondimensional velocity data, see Fig. 5. The exerimental (nondimensional) velocities are seen to be about 10% smaller than the fully nonlinear theoretical reference and the exonential rofile. The random wave series number 2 has the same nondimensional value of v 2 H s =2g as in series 1, and the range of the wave sloe of the selected wave events is in the same range in the two series. The velocity rofiles shown in Fig. 6 are rather close to the observations in series 1 and 4, aart from the velocity rofiles in the late art of series 2, i.e. at times s (marked in the figure by A) and s (marked by S). The velocity has then got an additional tilt: the nondimensional velocity near the crest has become larger, and the velocity below the mean water level smaller, than in the revious recordings. The waves are still rather stee waves in dee water. The estimated wave sloe in the last recording of series 2 is e ¼ 0:24 and the wavenumber times the water deth is kh ¼ 2:9; for examle. The relatively large value of kh indicates that the effect of a finite water deth cannot fully exlain the additional tilt. We note that the waves in series 2 are 40% longer than those in series 1 and 4. The wave velocity (grou velocity) is corresondingly 20% higher and the estimated Stokes drift 60% higher. Corresondingly, an estimated return velocity beneath the wave is 120% higher in series 2 than in series 1 and 4. An addition to the return flow due to a finite length of the tank is set u quicker and becomes stronger in series 2

9 J. Grue et al. / Alied Ocean Research 25 (2003) Fig. 6. Horizontal velocity rofiles below wave crest. Five wave events of series 2. (t 1 ¼ 124:17 s, e ¼ 0:26; kh ¼ 2:6; X), (t 2 ¼ 162:02 s, e ¼ 0:21; kh ¼ 2:6; *), (t 3 ¼ 184:45 s, e ¼ 0:29; kh ¼ 2:7; þ), (t 4 ¼ 226:95 s, e ¼ 0:34; kh ¼ 2:4 A), (t 5 ¼ 304:62 s, e ¼ 0:24; kh ¼ 2:9; S). Solid line: Velocity rofile in freak wave comutation with e ¼ 0:29: Dotted line: e ky : than in 1 and 4. With the PIV equiment we are able to erform recise measurements of the instantaneous velocity field. Recordings of long time series of the wave field using PIV is revented by limitations of the aaratus, however, and we are unable to measure the induced drift velocities in the wave tank using the equiment. We are thus left with estimates of a time averaged (or slowly varying) velocity field in the wave tank, as indicated above. We seculate that a relatively ronounced time averaged (slowly varying) return velocity is resonsible for the additional tilt of the velocity rofiles observed in the late art of the exeriments in series 2. The results from series 3 (Fig. 7) show the same tendency as in series 2 aart from the first event in the series (marked by the thin dots) where the local wave has rather small value of e ¼ 0:14 and moderate wavenumber kh ¼ 1:9: The measured nondimensional velocities early in the series are close to the theoretical references (fully nonlinear comutation and exonential rofile). An additional tilt aears in the velocity rofiles late in the runs, however. Some scatter in the data of the fourth wave event (at time t 4 ¼ 226:9 s)is seen. In series 5 (results not shown) the waves are dee water waves ðkh. 2:9Þ and the range of e is The data agree with the results from series 1 and 4, more or less. While some scatter in the data is noted, the sread seems not to be systematic. In series 6 (results not shown), the wave sloe is systematically smaller than in the other exerimental series ð0:17, e, 0:22Þ: The recordings early in the series are close to the theories. Late in the series the velocity rofiles become somewhat tilted Very stee wave events Results from all stee events in the random wave series, with e. 0:3; are lotted together in Fig. 8. The figure shows a relatively systematic aearance of the velocities in stee wave events in random wave fields. The measured rofiles above the mean water line, ut on nondimensional form, almost solely deend on the local estimated wavenumber and wave sloe, when the latter is large. An uer bound of the velocities is given by the fully nonlinear comutation and exonential rofile, ractically seaking. This conclusion is true also for events in late arts of the time series when there is evidence of an enhanced return flow due to the finite length of the wave tank. This conclusion also holds even if there is some (small) effect of a finite water deth (kh ¼ 2:4 in one of the runs). Fig. 7. Same as Fig. 6, but series 3. (t 1 ¼ 117:63 s, e ¼ 0:14; kh ¼ 1:9; X), (t 2 ¼ 166:35 s, e ¼ 0:24; kh ¼ 2:7; *), (t 3 ¼ 213:14 s, e ¼ 0:26; kh ¼ 2:7; þ), (t 4 ¼ 226:90 s, e ¼ 0:34; kh ¼ 3:7; A), (t 5 ¼ 323:57 s, e ¼ 0:23; kh ¼ 2:3; S). Fig. 8. Horizontal velocity rofiles below wave crest. All large wave events in the random waves series 1 6, with e. 0:3: Solid line: Velocity rofile in freak wave comutation with e ¼ 0:29: Dotted line: e ky : (Note the differences in horizontal scale.)

10 364 J. Grue et al. / Alied Ocean Research 25 (2003) The exonential rofile somewhat overestimates the velocities below mean water line Effects of a finite water deth A finite deth effect makes an imortant change of the velocity rofile below the wave crest, comared to dee water: near the crest of the wave the velocity becomes increased, and below the mean water line the velocity becomes reduced. Such tendencies are observed in our random wave recordings with 2:3, kh, 2:6 and e. 0:2: 4.4. Comments on exerimental error sources The strong collase of the measurements resented in Fig. 3 indicates the high accuracy of the exerimental rocedure. The relatively narrowbanded wave fields in the exeriments with the leading wave of a wave grou and the focusing waves fit to a slowly varying modeling of the wave field. An exerimental error may be quantified by the deviation from an average line through all exeriments in Fig. 3a c. The exeriments are very reeatable, with the velocity recordings showing a scatter being tyically lus/minus one grid oint (corresonding to 0.7 cm; there are 40 grid oints in each direction). Further, the PIV technique has a theoretical relative error that is about 2%. The wave elevation is measured using wire gauges, with a relative error less than 5%. While a high accuracy of the PIV recordings generally can be obtained close to a boundary, some scatter in the velocity recordings is observed near the free surface, in some of the runs. High quality velocimetry near the free surface using small field of views will increase accuracy and reduce eventual scatter in the data. We note, however, that the maximal ossible velocities induced by the waves are catured by the resent exerimental camaign. 5. Discussion 5.1. Comarison with other work Baldock et al. [7] measured the velocities in stee focusing waves in dee water using LDA. The data from the time record of the steeest wave (their figure 11, case D (c)) are: T ¼ 0:904 s and h m ¼ 7:49 cm, giving e ¼ 0:29; kh m ¼ 0:34; kh ¼ 3:2: The value of kh indicates that the waves can be considered as dee water waves. Their velocities [7, figure 13c] ut in nondimensional form fit excellent with the resent theoretical and exerimental results for dee water, see Fig. 3c. In the figure is also included the second-order model redictions from Ref. [7]. The second-order model shows a smaller velocity than the measurements. In the case studied by Baldock et al. the Wheeler stretching method [8] is quite close to the secondorder model (with e ¼ 0:29; kh m ¼ 0:34; results not shown). The Wheeler method significantly underredicts the kinematics of the larger wave, with e ¼ 0:46; kh m ¼ 0:62; as indicated in Fig. 3c. The wave data for the largest (focusing) wave in Kim et al. [5] are: T ¼ 1:36 s and h m ¼ 15 cm, giving e ¼ 0:28; kh m ¼ 0:34; kh ¼ 2:04: The value of kh indicates that the effect of a finite water deth is imortant in their measurements. The velocity data for their largest wave fit well with our random wave measurements for 2:3, kh, 2:6 and 0:7, u=e ffiffiffiffi g=k, 1:6 (results not shown). The large velocity at the crest observed by Kim et al., with nondimensional value u=e ffiffiffiffi g=k. 2:3; is not observed in the resent measurements. The data of the lunging wave measured by Skyner [9] are: T ¼ 1:03 s and h m ¼ 10:2 cm, giving e ¼ 0:3; kh m ¼ 0:36; kh ¼ 2:6: The measured horizontal velocity below crest is 2 m s 21 at maximum, i.e. u=e ffiffiffiffi g=k. 4: Such a high velocity is not observed in the resent dee water wave measurements. In the large scale FULWACK exeriment [2], the main urose was to obtain velocity measurements at locations relatively high above the mean sea level using current meters. These were located at 26, 16, 6 ft (8.2, 5, 1.9 m) above mean sea level. The largest observed seed at the to current meter was ft/s (6.5 m/s). The latform was in 270 ft (85 m) of water. For illustration, we assume for the moment that a wavenumber of k ¼ 0:03 m 21 is reresentative for the large scale measurements (the time record of the wave elevation was not given in the aer). The corresonding wave length then becomes 2=k ¼ 209 m, and kh ¼ 2:6: For the osition of the uer current meter this gives ky ¼ 0:03 8:2 ¼ 0:246: Emloying an exonential velocity rofile, i.e. u=e ffiffiffiffi g=k ¼ e ky ; we obtain e. 0:28: The resulting maximal elevation of the wave becomes 11 m and the estimated maximal fluid velocity at the crest becomes 7.04 m/s. Similar estimates can be given with smaller or larger k: 5.2. Fluid velocity relative to the wave velocity Kim et al. [5] measured a maximal fluid velocity being 64% of the estimated wave seed. This result may be comared with u=c in the resent measurements, where the nonlinear ffiffiffiffiffiffiffiffi wave celerity may be estimated by c ¼ v=k. 1 þ e 2 ffiffiffiffi ffiffiffiffiffiffiffiffi g=k : This means that u=c. ea= 1 þ e 2 ; where a is the nondimensional fluid velocity lotted in the figures. We obtain the following table of the maximal values of u=c: e kh m kh u=c Model, Sections 2.3 and Baldock et al. [7] Present exeriments Kim et al. [5] Skyner [9]

11 J. Grue et al. / Alied Ocean Research 25 (2003) For dee water waves, the resent model redictions give a value of u u to 40% of the wave seed. These comutations corresond directly to the measurements obtained in Ref. [7]. The resent dee water wave exeriments document a fluid velocity u to 75% of the wave seed c (and kh m ¼ 0:62). The increase in the maximal value of u=c; from 0.4 in the model redictions, to 0.75 in the exeriments, is in fact formidable. The latter is almost the double of the former. The lunging breaker measured by Skyner [9] has a fluid velocity that may even exceed the wave seed (u=c. 1:14; kh m. 0:36; kh ¼ 2:6) Exonential rofile In all figures we have included the nondimensional rofile, e ky where the wavenumber is determined according to the rocedure in Section 2.4. The exonential rofile is observed to be rather close to all the exerimental observations in the to of the wave when they are ut on nondimensional form. This is true rovided that kh is large and the fluid velocity due to a return flow beneath the waves is small. The exonential rofile gives relevant estimates for all observations in dee water and u=c u to 0.75, and kh m u to This is significantly beyond the level where the fully nonlinear model gives useful redictions. 6. Conclusion The kinematics of stee water wave events has been studied comaring PIV measurements in a wave tank and theoretical/numerical redictions using a fully nonlinear wave model. A total of 62 different stee wave events have been measured in the laboratory. Several generation mechanisms have been ursued. We have studied the leading unsteady art of a wave train, waves resulting from a focusing technique and events taking lace in random wave series. In the random wave exeriments each of a total of six series had three reetitions showing almost no scatter of the results. The Stokes drift and a corresonding return flow beneath the wave grous is inherent in all the exeriments and in the fully nonlinear comutations (where the latter is effective u to breaking). We have aimed at a general way to characterize the wave events. From the wave record (at a fixed oint) we identify the trough-to-trough eriod, T TT and the maximal elevation, h m of an individual stee wave event. The local wavenumber, k and an estimate of the local wave sloe, e are obtained from the equations v 2 =ðgkþ ¼1 þ e 2 ; kh m ¼ e þ 1 2 e 2 þ 1 2 e3 ; where v ¼ 2=T TT and g denotes the acceleration of gravity. The velocity fields are ut on nondimensional form dividing by e ffiffiffiffi g=k : The rocedure is documented in full in Section 2.4. Almost all of the exerimental waves can be characterized as dee water waves. A strong collase of the nondimensional velocity rofiles is observed for the 13 different large leading waves events of a wave train and the 19 focusing wave events. The LDA velocity measurements by Baldock et al. [7] ut on nondimensional form fit excellent with our velocity rofiles, both those obtained by PIV and those obtained by the fully nonlinear theory (Section 2.3). The kinematics in the random wave exeriments with a small return fluid velocity beneath the waves conforms with the kinematics in focusing waves and the leading wave of a wave grou. In some of the exeriments there is a weak effect of the finite deth of the fluid, with kh tyically in the range A larger velocity close to the wave crest and a smaller velocity below the mean water line than in dee water is then seen. The exeriments with the random wave fields are run so long that an additional return flow in the wave tank is induced. A return flow below the wave grou generally reduces the velocity below mean water level and is resonsible for an additional tilt of the velocity rofile, deending on the magnitude of the return fluid velocity. This may be determined by the integrated return flow divided by the water deth below the wave, see Section While the PIV technique records the instantaneous velocity field, we are unable to identify the return fluid velocity in the resent exeriments. Some scatter is observed in some of the random wave exeriments with a moderate wave sloe. This may be due to waves of different wavenumbers that are locally interacting. The milder wave events seem to be less clean than the larger waves where there is relatively little scatter in the results. Predictions using second-order model or extensions seem relevant if the sea state is moderate [15 17]. The fully nonlinear theoretical model is useful for the rather stee waves, inducing a fluid velocity u of u to 40% of the wave seed. This is recisely the same level as in the exeriments by Baldock et al. [7]. Both in the simulations and in the exeriments by Baldock et al. the maximal wave sloe was e ¼ 0:29: In the resent exeriments we have been able to ush the uer value of u=c (in dee water) u to 0.75, i.e. a fluid velocity u to 75% of the estimated hase seed of the wave. The estimated value of kh m is The focusing wave measurements [5] and [9] exhibit maximal u=c ¼ 0:64 and 1.14, resectively. The corresonding estimates of kh m are 0.33 and 0.36, resectively. A surrising result is that the exonential rofile e ky comares rather well with all measurements of the waves (ut on nondimensional form) which are in dee water and when the return velocity beneath the wave grou is small. The good comarison suggests that the exonential rofile is quite useful for obtaining estimates of the kinematics of stee waves. It comares well with all measurements resented here, where the main focus is the velocity rofile above mean water level, even for waves where the fluid velocity is u to 75% of the estimated wave seed. We note that the exonential rofile should be used in connection

12 366 J. Grue et al. / Alied Ocean Research 25 (2003) with the rocedure determining the local wave sloe and wavenumber outlined here. The Wheeler stretching method, widely used in engineering ractice, relaces the vertical coordinate y by y 0 ¼ hðy 2 hþ=ðh þ hþ: For large water deth this means y 0 ¼ y 2 h: It is easily seen that our results do not suort the Wheeler stretching method. In fact, for the steeest waves investigated here, the Wheeler method redicts a article velocity under crest being about one-half of the true velocity, see Fig. 3c. On this oint our results are closer to a velocity rofile that is linearly extraolated above the mean water line. Longridge et al. [6] comared velocities from LDA measurements of random wave series with theoretical models including linear extraolation, Wheeler stretching and a hybrid wave model. They found that the former was the suerior one for the range of wave sloe under investigation ðe ¼ 0:11 0:19Þ: For this range, the exansion e ky ¼ 1 þ ky is valid, indicating mathematically the usefulness of a linear extraolation of the velocity field above mean water line. This conclusion has been suorted by second-order comutations [24] comaring with irregular wave exeriments [4]. A linear extraolation underestimates the kinematics of stee wave events, however. The resent exeriments focusing on the velocity below crest, confirm the value of irrotational flow comutations of the entire velocity field below stee waves (which are now rather quick). Acknowledgements This work was conducted under the Strategic University Programme General Analysis of Realistic Ocean Waves funded by the Research Council of Norway. The technical assistance in the Hydrodynamics laboratory by Mr Arve Kvalheim and Mr Svein Vesterby are gratefully acknowledged. References [1] Buckley WH, Stavovy AB. Progress in the develoment of structural load criteria for extreme waves. Proceedings of the Extreme Loads Resonse Symosium. The Shi Structure Committee and the Society of Nav. Arch. and Mar. Engng. (SNAME), USA; [2] Forristall GZ. Kinematics in the crests of storm waves. Proceedings of the 20th Coastal Engineering Conference, Taiei, Taiwan; [3] Skjelbreia JE, Gudmestad OT, Ohmart RD, Berek G, Bolen ZK, Heideman JC, Sidsoe N, Tørum A. Wave kinematics in irregular waves. OMAE I-A, Offshore Technology; [4] Skjelbreia JE, Tørum A, Berek E, Gudmestad OT, Heideman JC, Sidsoe N. Laboratory measurements of regular and irregular wave kinematics. NorgesTekniske Universitetsbibliotek. Eske 604; [5] Kim CH, Randall E, Boo SY, Krafft MJ. Kinematics of 2-D transient water waves using Laser Doler Anemometry. J Waterway, Port, Coastal Ocean Engng 1992;118(2): [6] Longridge JK, Randall RE, Zhang J. 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A fast method for fully nonlinear water-wave comutations. J Fluid Mech 2001;447: [21] Clamond D, Grue J. On efficient numerical simulations of freak waves. Proceedings of the 11th International Offshore and Polar Engineering Conference and Exhibition, Stavanger; June 17 22, ISOPE [22] Clamond D, Grue J. Interaction between enveloe solitons as a model for freak wave formations. CR Acad Sci Paris 2002;330: [23] Taylor PH. On the kinematics of large ocean waves. Proc Behav Offshore Struct Conf (Lond) 1992;1: [24] Stansberg CT, Gudmestad OT. Non-linear random wave kinematics models verified against measurements in stee waves. Proc Offshore Mech Arctic Engng Conf (OMAE. Florence) 1996; [25] Moe G, Arntsen Ø, Gjøsund SH. Wave kinematics based on a Lagrangian formulation. In: Zhang J, editor. Proceedings of the Ocean Wave Kinematics, Dynamics and Loads on Structures. ASCE Symosium, Houston,TX; [26] Grue J, Bjørshol G, Strand Ø. Higher harmonic wave exciting forces on a vertical cylinder. 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