Spatial Evolution of Water Surface Waves: Numerical Simulation and Experiment of Bichromatic Waves

Transcription

1 Proceedngs of the Eleventh (21) nternatonal Offshore and Polar Engneerng Conference Stavanger, Norway, June 17-22, 21 Copyrght 21 by The nternatonal Socety of Offshore and Polar Engneers SBN (Set); SBN O(VoL 11); SSN (Set) Spatal Evoluton of Water Surface Waves: Numercal Smulaton and Experment of Bchromatc Waves Karsten Trulsen SNTEF Appled Mathematcs Oslo, Norway Carl Trygve Stansberg Marntek Trondhem, Norway ABSTRACT The modfed nonlnear Schr6dnger (MNLS) equaton for spatal evoluton of weakly nonlnear water surface waves s shown to yeld good comparsons wth expermental measurements of bchromatc waves n a long tank. Whle lnear theory does not predct nether the phase velocty nor the evoluton of the envelope well, the cubc nonlnear Schr6dnger (NLS) equaton mproves the predcton of the phase velocty but not the modulaton of the envelope. The MNLS equaton predcts both the evoluton of ndvdual wave crests and the modulaton of the envelope over longer fetch, and thus permts accurate forecastng of ndvdual ocean wave crests over a fetch of several tens of wavelengths. KEY WORDS: wave groups, bchromatc waves, nonlnear Schr6dnger equaton NTRODUCTON Wave groupng s a promnent feature of ocean waves, frequently dscussed n the lterature. t may be partly responsble for the generaton of freak waves. The smplest realzaton of wave groups s the bchromatc wave, acheved by mxng two monochromatc waves. n the present paper we use experments of bchromatc waves as a benchmark to assess the capablty of three dfferent smulaton models to descrbe wave group evoluton. The three models are the lnear wave equaton, the cubc nonlnear Schr~dnger (NLS) equaton and the modfed nonlnear SchrSdnger (MNLS) equaton. t s common to stress the mportance of tme-doman smulaton, as opposed to frequency-doman smulaton, of nonlnear ocean waves. However, conventonal methods for measurng waves n the laboratory and the feld yeld tme seres at selected spatal locatons. Space evoluton s mpled between the selected ponts. For such cases a tme-doman smulator s lkely not very useful, or at best qute dffcult to ntalze. A better approach s to nterchange the role of space and tme n the evoluton equaton to obtan a genune space-doman smulator. The nonlnear SchrSdnger equaton and ts hgher order modfcatons are partcularly well-suted for ths purpose. Lo ~ Me (1985) frst presented comparsons between experments and the space evoluton predcted by the MNLS equaton, and obtaned good results. Recently the MNLS equaton was enhanced wth exact lnear dsperson (Trulsen et al. 2) for better bandwdth resoluton for applcaton to realstc ocean wave spectra. Ths approach s based on the assumpton that the spectrum to leadng order of approxmaton s narrow-banded. The remanng part of the spectrum s reconstructed only to the extent that t s nonlnearly forced by, and thus coherent wth, the lnear waves near the spectral peak. Specal care must be taken for proper ntalzaton to dstngush between lnear free waves and nonlnearly forced waves. To ths end we have developed an teratve technque by whch the extracted spectrum of lnear free waves s refned untl exact reconstructon of the measured complex spectrum has been acheved wthn the bandpass regon. The natural spatal scale of nonlnear modulaton s ~/= e2kcx, where e = kcac s the wave steepness, kc and ac are characterstc scales for wavenumher and ampltude, and x s the fetch. The present smulaton results of the MNLS equaton suggest that predcton of the evoluton of ndvdual wave crests s good at least up to ~/-- 3, whle t becomes poor for ~/> 5. f we assume e = for typcal ocean waves, then ~} = 5 roughly corresponds to 8 wavelengths. For 12 s waves on deep water, correspondng to 22 m wavelength, that could mply up to half an hour warnng for ndvdual wave crests for propagaton over 16 km n long-crested seas. Proper verfcaton should be done aganst expermental data for short-crested seas. Smlar work wth the cubc nonlnear Schr~dnger (NLS) equaton was done by Shemer et al. (1998) for deep water and wth the Korteweg de-vres equaton for shallow water by Kt et al. (2). The Zakharov equaton, whch n ts orgnal form s a tme-doman equaton, has been dscretzed for applcaton to measurements (Rasmussen ~ Stassne 1999); recently t was cast as a space doman smulator for one horzontal dmenson by She- 71

2 mer et al. (21). THE EXPERMENT The experments were carred out n the 26 m long and 1.5 m wde towng tank at Marntek. Waves were generated by a horzontally double-hnged flap-type wave maker. The depth of the tank s 1 m for the frst 8 m closest to the wave maker, and 5 m elsewhere. A vertcal bottom jump connects the two tank parts. A slopng beach s located at the far end of the tank opposte the wave maker. The wave elevaton was measured by wave staffs at dfferent locatons smultaneously downstream the tank, see fgure 1. wavemaker beach. Y, / Recently, Trulsen et al. (2) explaned how the MNLS equaton can be enhanced wth exact lnear dsperson by ntroducng a pseudo-dfferental operator for the lnear part. n two horzontal dmensons t reads OB " 3 _.2OB 1BeOB* O---t- +L(O~'Ou)B+ 2 B2B+ ~l~l ~ + 4- x +l~--~x B = at z=, (3) 1 2 Oz -- 2 ~B at z =, (4) V2@= for -h<z<, (5) x (m) Fgure 1: Sketch of the wave tank wth the locatons of the wave staffs. The tme seres measured by probe 1 at 9.3 m s used n ths paper for ntalzaton of the numercal smulatons. Probe 2 s at 4 m and s.5 m n front of probe 3. Probes 6 and 8 are.4 m ahead of and behnd probe 7 at 16 m. Probe 9 s 1.5 m from the tank wall and serves to assess the mportance of transversal modulatons. We consder two experments wth bchromatc waves (test 6 and test 61) whch were done accordng to the followng specf- Catons: Test Wave perods Tlsl T2sl Wave heghts H1 [m] H2 [m] Further detals of the experment have been reported n Stansberg (1993, 1995, 1998). MATHEMATCAL MODEL FOR SPACE-DOMAN SM- ULATON Startng from the nvscd equatons for potental flow, normalzed by the characterstc wavenumber k~ and frequency w, and assumng constant depth h whch s great n comparson wth the wavelength k~h >> 1, we make an assumpton that the velocty potental ~b and surface dsplacement ~ of the wave feld can be expanded n harmonc expansons = +5 1 (Ae(X_t)+~ + A2e21(z_t)+2 z =~+~ +Aze s(~-t)+3~ c.c.~ (1) ] 1 [ ~Be (~-t) + B2e 21(z-t) + B3e a(~-t) c.c.). \ (2) Here (z, V) and z are horzontal and vertcal coordnates, t s tme, and ~ are mean feld varables, and A, An, B, Bn are complex harmonc ampltudes. The complex conjugate s denoted by "c.c." where the pseudo-dfferental operator L s -~ = at z=-h, (6) L(O~, v) = { [(1 - ~) 2 - ~],/4 _ 1}. (7) These equatons can be nverted wth respect to space and tme to yeld a space-doman formulaton 13 + (O~,O~,)B + B2B - 8lB2-~ -. - x -4-~B-- at z=, (8) o$ O 2 O---~= ~[B[ at z=, (9) V~ = for -h<z<, (1) ---~= at z=-h. (11) Here we have used the fact that /x = -2 /t to the leadng order. Ths transformaton should also be reflected n the Laplacan of the nduced velocty potental n (1). The pseudodfferental operator becomes r-(t,y) = - { [(1 + t) x/2-1}. (12) By expandng the lnear pseudo-dfferental operators L or L: n power seres expansons and truncatng at approprate orders, we recover the MNLS equaton of Dysthe (1979) and the broader bandwdth equaton of Trulsen ~ Dysthe (1996). Furthermore by truncatng the nonlnear part to retan only the leadng cubc nonlnear term, we recover the standard cubc nonlnear SchrSdnger equaton. We remark that n one horzontal dmenson (x), the operator s algebrac. n ths paper we consder the one-dmensonal (x) lmt of (8)-(12). The MNLS equaton becomes OB 2B 2B 2B 2B* Ox b --5~ + -~-~- + lb2 B -- 8[B -~ - 2B --~ -4-~-~B= at z=. (13) U# 72

3 The cubc nonlnear (NLS) equaton s The lnear equaton s OB OB.2B Ox + 2-~- + 1-~ + ]S[2B =. (14) OB OB.2 B -~ + 2-oy +,-~ =. (15) The reconstructon of the surface dsplacement (2) s acheved by the formulas ~= 6) Ot ' 1 B 2 OB B2 = + Bm-, (17) t s n general not possble to deduce what porton of the power spectrum s due to free or bound waves based on a tme seres from a sngle pont. A good approxmaton s to construct /~ by bandpassng the complex spectrum ~ around a central frequency. To ths end we determne the characterstc frequency wc from the mean of the dmensonal frequency power spectrum 27rMo Ey wj [~J[ 2 ~o = - - ~ (~1) T y~q [~[2 ' such that M s an ntegral number of central wave oscllatons n the computatonal doman. The correspondng characterstc wavenumber kc s computed from the lnear dsperson relaton. The complex spectrum of the frst harmonc B s frst assgned by bandpassng Mbp components of the desred complex spectrum ~j centered around we, and S(~,~) = 2cje-lZ~(~,wj+Mo) for Jl ~ -'~-"--~P, (22) NUMERCAL MPLEMENTATON B3 = 3Ba. (18) The present mplementaton s based on mposng perodc boundary condtons n tme, whch s approprate for the expermental data under consderaton. Lettng B be represented by a sutable number of modes M, the Fourer transform par for B s defned by M--X [~(x,wj) = M E B(x'tm)e~t'~' m=o (19) s(~,t~) = ~ b(~,~)e -~, V/ M where cy are adjustable coeffcents ntally set to unty. As far as/} s concerned, wj s a modulaton frequency relatve to the characterstc frequency we. The complex coeffcents cj are adjusted n an teratve manner to compensate for the hgher-order nonlnear modfcaton of the measured wave spectrum (16)-(18). f the desred spectrum ^ s ~j, suppose that after an teraton the reconstructon of the spectrum s ~j and the reconstructon of the lnear frst-harmonc part of the spectrum s ~x,j, then the new terate for cj s ~ - 5,~ (23) cj=l ~j The teraton scheme s stopped when the adjustment coeffcents have converged, typcally after 1-2 teratons. where wj = 2rj/T, tm = mt/m, and T s the length of the perodc doman. The numercal mplementaton has been documented elsewhere (Lo &: Me 1985; Trulsen & Dysthe 1997). The dfferental equaton s solved by an operator splttng method. The nonlnear part s ntegrated by a fnte dfference method n physcal space, whle the lnear dspersve part s solved exactly n Fourer space. n partcular, we remark that the full pseudo-dfferental operator becomes algebrac n Fourer space, and thus ntegraton of the exact lnear dspersve part n two horzontal dmensons can be done as fast and easly as for any of the truncated equatons. NTALZATON The surface dsplacement ~(x,t) s measured at a fxed poston x wth a samplng perod r, provdng a tme seres ~(x, t,~) for t, = nt, r~ = O, 1, 2,..., N- 1 over a total tme nterval T = N~-. The Fourer transform par for the surface dsplacement s where wj = 27tlT. N--1.=o (2) C(~,t.) = ~ ~(~,~)e -'~'~, N N COMPARSON BETWEEN PERMENT SMULATON AND EX- Expermental test 6 s perodc wth perod 39.9 s. We use a computatonal doman of length s, correspondng to 7 perods, after skppng the frst 19.7 s of startup. The nondmensonal depth s kch = 1 for the frst 8 m and kch = 5 for the rest of the tank, however we here present smulatons usng kch = 1 for the entre tank. Smulatons wth kch = 5 revealed only nsgnfcant modfcatons for large fetch, thus we beleve that as far as ths comparson between experment and smulatons s concerned, the effect of the jump at 8 m s not mportant. The tme seres measured at wave staff 1 s used for ntalzaton. Here the transent effects of startup do not occur n the computatonal doman. At successve wave staffs, the transent effects of startup propagate nto the computatonal doman, but are not accounted for n the numercal smulaton. We present results from the last perod n the computatonal doman whch remans unaffected by transent effects of startup for the duraton of the smulaton. Measurements and ntalzaton at staff 1 s shown n fgure 2. Lnear wave theory at staffs 2, 4 and 5 are shown n fgures 3-5. Lnear theory underpredcts both the phase and group veloctes observed n the experments. Lnear theory also does not account for the change n shape of the wave group. NLS smulaton results at staffs 2, 4 and 5 are shown n fgures 6-8. The NLS equaton accounts for a nonlnear ncrease 73

4 n phase velocty, and yelds good agreement wth the observed phase velocty n the experment. The NLS equaton does not account for the nonlnear ncrease n group velocty. The NLS equaton does account for the nonlnear ncrease n ampltude of the group, but does not capture the asymmetrc forward-leanng evoluton seen n the experments. MNLS smulaton results at staffs 2, 4, 5, 7 and 1 are shown n fgures The MNLS equaton accounts for nonlnear ncrease n both the phase and group veloctes, n good agreement wth the expermental observatons. The nonlnear ncrease n ampltude of the group and the asymmetrc forward-leanng evoluton of the group seen n the experments are also captured. Experment 61 was done for shorter wavelength and larger steepness, yeldng an effectvely much longer tank. The waves are perodc wth perod s. We use a computatonal doman of length s correspondng to 1 perods, after skppng the frst 27.5 s of the tme seres. The nondmensonal depth s kch = 4 for the frst 8 m and kch = 2 for the rest of the tank. These depths are effectvely nfnte as far as experment 61 s concerned. The tme seres measured at wave staff 1 s used for ntalzaton and s shown n fgure 14. MNLS smulaton results at staffs 2, 4 and 5 are shown n fgures n fgure 15 we observe that the qualtatve features of group splttng are well captured. The last fgure 17 reveals that the smulaton results of the MNLS equaton become unrelable for large fetch. CONCLUSON We have shown that the MNLS equaton can be used to predct the evoluton of ndvdual long wave crests at least up to the dmensonless scale for evoluton ~ = ~2kcx The predcton becomes poor for ~ ) 5, and s unrelable for ~7 ) 8. f we set = for typcal ocean waves, then ~ = 5 roughly corresponds to 8 wavelengths. For 12 s waves on deep water, correspondng to a wavelength of 22 m, that could mply up to half an hour warnng for ndvdual wave crests at a dstance of up to 16 km for a long-crested sea. On the other hand, the NLS equaton and the lnear theory are not able to predct the evoluton well even up to ~ = 1. The characterstc qualtatve features of bchromatc wave evoluton, e.g. nonlnear ncrease n phase and group veloctes, asymmetrc forward-leanng evoluton of ntally symmetrc groups, and group splttng, are captured by the MNLS equaton. On the other hand, the standard cubc nonlnear SchrSdnger (NLS) equaton only accounts for the nonlnear ncrease n the phase velocty. We conclude that the hgher order nonlneartes orgnally found by Dysthe (1979), are essental to explan nonlnear ocean wave evoluton, even over short fetch. These experments and smulatons were done for long-crested waves. t s necessary to perform experments and smulatons for short-crested waves to assess f forecastng of ndvdual wave crests can be done n a realstc short-crested sea. However, for long-crested sea states, we antcpate that ths model s capable of accurately forecastng ndvdual wave crests over a fetch of several tens of wavelengths. REFERENCES Dysthe, K. B. (1979). Note on a modfcaton to the nonlnear SchrSdnger equaton for applcaton to deep water waves. Proc. R. Soc. Lond. A 369, Kt, E., Shemer, L., Pelnovsky, E., Talpova, T., Etan, O. ~z Jao, H.-Y. (2). Nonlnear wave group evoluton n shallow water. J. Waterway, Port, Coastal and Ocean Engneerng 126, Lo, E. 8z Me, C. C. (1985). A numercal study of water-wave modulaton based on a hgher-order nonlnear SchrSdnger equaton. J. Flud Mech. 15, Rasmussen, J. H. &: Stassne, M. (1999). Dscretzaton of Zakharov's equaton. Eur. J. Mech. B/Fluds 18, Shemer, L., Kt, E., Jao, H.-Y. &: Etan, O. (1998). Experments on nonlnear wave groups n ntermedate water depth. J. Waterway, Port, Coastal and Ocean Engneerng 124, Shemer, L., Jao, H. Y., Kt, E. & Agnon, Y. (21). Evoluton of a nonlnear wave feld along a tank: experments and numercal smulatons based on the spatal Zakharov equaton. J. Flud Mech. 427, Stansberg, C. T. (1993). Propagaton-dependent spatal varatons observed n wavetrans generated n a long wave tank. Data report. Techncal Report MT49 A93-176, Marntek. Stansberg, C. T. (1995). Spatally developng nstabltes observed n expermental bchromatc wave trans. n A. J. Grass (Ed.), 26th AHR Congress (HYDRA 2), volume 3 (pp ): Thomas Telford. Stansberg, C. T. (1998). On the nonlnear behavour of ocean wave groups. n Proe. Thrd nternatonal Symposum on Ocean Wave Measurement and Analyss -- WAVES'97 (ASCE), volume 2 (pp ). Trulsen, K. & Dysthe, K. B. (1996). A modfed nonlnear SchrSdnger equaton for broader bandwdth gravty waves on deep water. Wave Moton 24, Trulsen, K. & Dysthe, K. B. (1997). Frequency downshft n three-dmensonal wave trans n a deep basn. J. Flud Mech. 352, Trulsen, K., Klakhandler,., Dysthe, K. B. &: Velarde, M. G. (2). On weakly nonlnear modulaton of waves on deep water. Phys. Fluds 12, ACKNOWLEDGMENTS Ths work has been supported by the Norwegan Research Councl through the project "Modelng of extreme ocean waves and ocean wave clmate on mesoscale" (139177/431) and by grants from Norsk Hydro and Statol (ANS2571). The experment was funded by the Norwegan Research Councl. 74

5 t ~ tll~ 11 t tl {... " Fgure 2: Test 6, wave staff 1 at 9.3 m, r/ --, used for ntalzaton: --, experment and all wave theores lnear, NLS, MNLS). Fgure 5: Test 6, staff 5 at 12 m, ~ : --, experment; - -, lnear.! Fgure 3: Test 6, staff 2 at 4 m, ~/--.4: --, experment; --, lnear. -4) Fgure 6: Test 6, staff 2 at 4 m, ~ --.4: --, experment; --, NLS.! \ o ' }, - ~ ~ " " ' ~ '~ ~ ' ~.... '' '] - ~! ~ ~,l ]' ', t', ~, ',,, t j f' ' - 4) f Fgure 4: Test 6, staff 4 at 8 m, ~ =.9: --, experment; --, lnear. Fgure 7: Test 6, staff 4 at 8 m, r/--.9: --, experment; - -, NLS. 75

6 {).3 o.2!,,,.~ - ' ' l ' X ' - v Fgure 8: Test 6, staff 5 at 12 m, rl = 1.4: --, experment;, NLS. Fgure 11: Test 6, staff 5 at 12 m, rl = 1.4: --, experment; --,3! ++ +,.., ++ t - - v Fgure 9: Test 6, staff 2 at 4 m, r 1 =.4: --, experment; Fgure 12: Test 6, staff 7 at 16 m, rl = 1.9: --, experment; +. ; - -4).1 u -.. -, t(s) Fgure 1: Test 6, staff 4 at 8 m, r/=.9: --, experment; Fgure 13: Test 6, staff 1 at 2 m, ~ = 2.4: --, experment;--, MNLS. 76

7 o C,6.-.(.) ,$6 t(s) Fgure 14: Test 61, staff 1 at 9.3 m, ~/=, used for ntalzaton: --, experment;- -.8!.8 + v,.j, o t #) h p,,, AA!... l!t ; llt ().OJ l (s) Fgure 15: Test 61, staff 2 at 4 m, 7/= 2.5:, -- experment; Fgure 17: Test 61, staff 5 at 12 m, ~ : --, experment; --.8 A ,~,,~l A >_,,~,~1.,, ) ',! ~. ) ' ', / t! _ -,6 -,8 44O Fgure 16: Test 61, staff 4 at 8 m, ~/= 5.7: --, experment; 77