EXPERIMENTAL STUDY OF BREAKING WAVES OVER A SHOAL Aru Chawla, H. Tuba Ozka-Haller ad James T. Kirb ABSTRACT: The aim of this paper is to stud the trasformatio of irregular directioal waves over a circular shoal. A experimetal stud has bee carried out. The resultig has bee used to test the accurac of a existig refractio-diractio. Model to comparisos have bee carried out for the etire basi regio icludig about shoal diameters dowwave of the shoal, with satisfactor results. Several phsical processes have bee idetied which lead to possible disparities i the comparisos. INTRODUCTION Wave ig of radom waves over a varig bathmetr is a subject of cosiderable importace to coastal egieers. The developmet of the classical mild slope equatio b Berkho (97) allowed coastal egieers to stud the combied eects of refractio ad diractio. A wide famil of equatios have bee derived from the mild slope equatio to icrease the accurac ad the speed of the s. Oe such set of equatios are the parabolic equatios, rst derived for ocea waves b Radder (979), which have gaied popularit because of their speed of computatio, eve though the have a xed directio of propagatio ad a limited rage of agles from the assumed propagatio directio over which the are valid. Noliear formulatios of parabolic equatios (Kirb ad Dalrmple 98) have bee foud to give more accurate results tha the liear mild slope equatio (Kirb ad Dalrmple 984). The limitatio i the rage of agles has bee relaxed usig Pade approximats (Booij, 98). Although relativel accurate parabolic s have bee developed to stud the evolutio of waves over a irregular bottom, all these s have bee derived for moochromatic waves ol. Coastal egieers have traditioall approximated the oshore irregular sea states b represetative moochromatic waves i order to use these s to make predictios. However, ivestigators such as Goda (985) (usig a aaltical approach), Vicet ad Briggs (989) (b coductig a experimetal stud) ad Pachag et al. (99) (usig a umerical approach) have show that such a approximatio ma result i large errors due to vast dissimilarities i the refractio-diractio patters of the two Ceter for Applied Coastal Research, Departmet of Civil ad Evirometal Egieerig, Uiversit of Delaware, Newark, DE 976, USA. Correspodece e-mail: cwla@coastal.udel.edu Chawla, Ozka & Kirb
.5 liear stokes H=Ho 4 6 8 4 6 8 Moochromatic waves 4 6 8 4 6 8 Radom waves Figure : Wave height distributio behid a shoal for a moochromatic wave ad a directioal sea state wave elds. Our ow experimets corm this ad Figure shows the vast diereces i the wave height distributio alog a trasect (trasect D-D i Figure ) behid a submerged shoal. Recetl, methods for computig the evolutio characteristics of a directioal spectral sea state usig parabolic s for moochromatic waves have bee developed. Pachag et al. (99), Grassa (99) ad Izumia ad Horikawa (987) have developed s usig a spectral calculatio method which cosists of discretizig the oshore spectrum ito idividual moochromatic directioal compoets, determiig the wave trasformatios of each compoet with the help of moochromatic wave s, ad the assemblig the wave compoets b liear superpositio at the respective grid poits i the domai to obtai the statistical characteristics of the spectrum at those poits. I this paper a umerical which has bee developed usig the parabolic formulatio of Kirb (986a), is tested agaist for a rage of breakig wave coditios. Experimetal stud of radom directioal waves breakig over a submerged circular shoal has bee carried out for two dieret directioal spreadigs ad eerg variaces to stud their eects o wave height distributio. Extesive surface elevatio measuremets have bee made o top of ad aroud the shoal, ad some aspects of the frequec spectra have bee looked ito. NUMERICAL MODEL The parabolic for spectral wave coditios used here simulates the evolutio of directioal radom waves i the earshore zoe. The predicts the eects of refractio, diractio, shoalig ad breakig. Therefore, the is particularl applicable to regios where a icomig radom sea propagates over complicated bathmetr towards shore. The bathmetr ma iclude a shoal formatio at the mouth of a ilet or estuar, where refractio, diractio, shoalig ad depth-limited breakig will be simultaeousl importat. The requires the specicatio of the icomig directioal radom sea at the oshore boudar. The radom sea is represeted b a two-dimesioal spectrum which is discretized ito wave compoets, resultig i wave compoets of amplitude A with associated frequec f ad agle of icidece to Chawla, Ozka & Kirb
6 5 4 9 O BEACH A B B C C 7 8 D D E E 9. F F 8. 5.. G G 7.9 9.8 4 A 6. 5..8 X Y WAVEMAKER 8. Figure : Schematic diagram of the experimetal setup the assumed propagatio directio, herei called x directio. The water surface elevatio ca be described i terms of these discrete wave compoets. It is assumed that the water surface elevatio is periodic i time ad that the spatial depedec ca be split ito a fast-varig phase ad a slow-varig amplitude. X ( ) X A(x; ; f; ) (x; ; t) = e i + c:c: () allf all where f is the frequec, is the directio of a idividual wave compoet ad Z = kdx?!t () The evolutio of these idividual wave compoets is computed simultaeousl at each forward step i the assumed wave propagatio directio usig a moochromatic wave. Therefore, after each forward step it is possible to determie statistical quatities at that row before takig aother step forward. These quatities are icorporated ito a statistical wave breakig (Thorto ad Guza, 98) which has bee added to the moochromatic wave. The refractio, diractio ad shoalig of the discrete wave compoets is assumed to be govered b the parabolic approximatio to the mild slope equatio derived b Berkho (97). To miimize the restrictios placed o the rage of allowed wave agles with respect to the assumed wave directio, the procedure derived b Booij (98) is used, eablig the to hadle wave directio up to about 45 from the x directio. The also has the abilit to hadle strog currets b usig the formulatio of the mild slope equatio icludig the iuece of currets derived b Kirb (986a). The goverig equatio for the wave compoet is: Chawla, Ozka & Kirb
Sf(cms sec) 5 Test Test 4 Test 5 Test 6 S 4.5 4.5 f(hz) frequec spectra..4.6.8..4.6.8 6 4 4 6 directioal spectra Figure : Desig frequec ad directioal spectra for the 4 directioal sea states Table : Test particulars for the radom wave experimets Test o. Hs(m) T p (sec) m Rage.9.7 5 4.56.7 45 5..7 5 6.49.7 45 + (C g + U)(A ) x? V (A ) + i( k? ak )(C g + U)A ( Cg + U V )? A + i " UV x A # + " UV 8 <? i : x 8 < " (CCg : )? V A +?b k + b (i! U + A x " (CCg )? V A? (! U A x A x # x A + i V # 9 = +! U x A # 9 = ; + A + i V? UV " (CCg )? V A A A! x 9 = ; ;? i k b f(! V ) + (! U) x g ) + ik! U(a? ) x A A x # = () where U ad V are the currets i the x ad directios, is a dissipatio coeciet for wave breakig, k is a represetative wave umber correspodig 4 Chawla, Ozka & Kirb
x(m) 9 8 7. 6. 5 4.5. 6 7 8 9 Figure 4: Refractio diagram at peak frequec, f p bathmetr. = :7Hz for the give to the peak frequec, ad The coeciets =!? k U; = (k ) x k = a? b; = + a? b; + (k ((CC g )? U )) x k ((CC g )? U ) ; = a? b k k : (4) a = ; a =?:75; b =?:5 (5) recover the Pade approximat of Booij(98). The statistical iformatio obtaied after each step i the parabolic scheme is used to costruct a for the dissipatio of eerg due to breakig.to determie the eerg dissipatio, a simple b Thorto ad Guza (98) is used. The eerg dissipatio is built ito the equatio usig a additioal breakig term A i (), so that it is uecessar to have a criterio for turig breakig o or o. The coeciet is give b = p 4 fb 4 h 5 H5 rms : (6) 5 Chawla, Ozka & Kirb
4 6 8 4 6 Test x(m) 4 6 8 4 6 Test 4 x(m) 4 6 8 4 6 Test 5 x(m) 4 6 8 4 6 Test 6 x(m) Figure 5: Sigicat wave height distributio alog trasect A-A where h is the local water depth ad f is a represetative frequec for the frequec spectrum ad is chose to be the peak frequec. B ad are costats ad are chose to be equal to ad :6, respectivel (Mase ad Kirb, 99). H rms is the root-mea-square wave height, ad is obtaied as a statistical quatit from the wave, H rms (x; ) = vu u tx N j A(x; ) j (7) = The dissipatio of Thorto ad Guza (98) was origiall derived assumig that the waves cotiue breakig oce the have started, ad has bee validated for waves breakig o a mootoic beach. Though we ca see from (6) that the dissipatio term is articiall reduced with icrease i local water depth h we are ot certai if this correctl simulates the reformig of waves with icreased water depth. Also, o modicatios have bee made i the dissipatio to accout for directioal eects. Ol chage i eerg ux i the x directio is cosidered, ad eerg ux i the directio does ot take part i the dissipatio. EXPERIMENTAL SETUP The experimets were coducted at the Ceter for Applied Coastal Research, Uiversit of Delaware. The wave basi is approximatel 8:m log ad 8:m 6 Chawla, Ozka & Kirb
4 6 8 4 6 8 Test 4 6 8 4 6 8 Test 4 4 6 8 4 6 8 Test 5 4 6 8 4 6 8 Test 6 Figure 6: Sigicat wave height distributio alog trasect F-F wide. It has a three-dimesioal wavemaker at oe ed, cosistig of 4 ap tpe paddles which creates the desired wave eld. The bottom is at except for a circular shoal i the ceter, ad a stoe beach at the far ed miimizes the reectios. A schematic view of the experimetal laout, together with the gage locatios is give i Figure. A total of te capacitace wave gages were used i the experimet, of which ie were placed o a arra. This arra was the placed at fourtee dieret positios (deoted b thick lies i Figure ) to obtai a total of 6 measurig poits aroud the shoal. Depedig upo their orietatio, oe or more arra positios form a trasect alog which comparisos are made with the umerical. There is oe logitudial trasect (A-A) goig over the top of the shoal ad six trasverse trasects (B-B, C-C, D-D, E-E, F-F ad G-G) behid ad o top of the shoal (see Figure ). The circular shoal has a diameter of 5:m ad a maximum height of 7cm. Geometricall it is the top portio of a circular sphere of radius 9:m. The ceter of the shoal is placed at x = 5m ad = 8:98m. The equatio for the perimeter of the shoal is give b (x? 5) + (? 8:98) = (:57) (8) ad for the bathmetr is give b q z =?h + 8:8? (x? 5)? (? 8:98)? 8:7 (9) 7 Chawla, Ozka & Kirb
4 6 8 4 6 8 Test 4 6 8 4 6 8 Test 4 4 6 8 4 6 8 Test 5 4 6 8 4 6 8 Test 6 Figure 7: Sigicat wave height distributio alog trasect E-E where h is the water depth awa from the shoal. Four dieret directioal sea test coditios (Test, Test 4, Test 5 ad Test 6) were ru with a TMA spreadig fuctio (Bouws et al., 985) i frequec, ad a wrapped ormal directioal spreadig fuctio (Borgma, 984) i directio. The water depth awa from the shoal (h i (9)) was 4cm, ad the water depth o top of the shoal was cm. All four tests had similar frequec spreadigs except that the eerg variace i Tests ad 4 were lower, ad the frequec spectra for the four test cases is give i Figure. I all the four cases the waves were breakig o top of the shoal, with more waves breakig for Tests 5 ad 6. Two dieret directioal spreadigs were used (Figure ), with the mea agle ormal to the wavemaker ( m = ). Tests ad 5 have a arrow directioal spread ( ), while Tests 4 ad 6 have a broad directioal spread (45 ). The iitial sigicat wave height (Hs), peak period (T p ), mea agle ( m ) ad the rage of directioal spreadig for the four dieret test cases are give i Table. Data was collected at a samplig rate of 5Hz for 655 secods (768 sample poits) at all the gages. DATA TO MODEL COMPARISONS Sigicat wave height iformatio was obtaied from the usig a zeroupcrossig method, while from the it was obtaied from the statistics assumig a Raleigh wave height distributio. Reectios from the beach at the 8 Chawla, Ozka & Kirb
4 6 8 4 6 8 Test 4 6 8 4 6 8 Test 4 4 6 8 4 6 8 Test 5 4 6 8 4 6 8 Test 6 Figure 8: Sigicat wave height distributio alog trasect D-D far ed of the basi (see Figure ) are a matter of cocer but have bee igored here sice the reected wave eld could ot be separated from the icidet wave eld. I each case the iput frequec spectrum to the was directl measured from the wave. The iput directioal spectrum was take to be the same directioal spreadig fuctio used to geerate icidet waves for the respective test cases (Figure ). A wave refractio patter for the peak frequec (Figure 4) shows that the focusig is quite severe o top of the shoal, ad that some of the wave ras are movig at agles greater tha 9. Sice the ca predict wave coditios accuratel ol withi a rage of wave agles of 45, some discrepacies betwee ad results are expected i this regio. For each test case, all sigicat wave height comparisos have bee ormalized b the respective iitial sigicat wave height (Hs) give i Table. Figure 5 gives the wave height comparisos alog trasect A-A. The teds to overpredict the wave height distributio ear the regio of focus. This is probabl due to the severe focusig i this regio (Figure 4), which the caot properl simulate. Sice the focusig is takig place iside the surf zoe, aother probable cause for the discrepac could be the limitatios of the breakig, ad a dieret breakig might give more accurate results. Comparisos alog the six trasverse trasects are show i Figures 6?. I all the cases the predicts large wave heights at the side walls. This is because the o ux boudar coditio at the side wall causes the waves to 9 Chawla, Ozka & Kirb
4 6 8 4 6 8 Test 4 6 8 4 6 8 Test 4 4 6 8 4 6 8 Test 5 4 6 8 4 6 8 Test 6 Figure 9: Sigicat wave height distributio alog trasect C-C Chawla, Ozka & Kirb
4 6 8 4 6 8 Test 4 6 8 4 6 8 Test 4 4 6 8 4 6 8 Test 5 4 6 8 4 6 8 Test 6 Figure : Sigicat wave height distributio alog trasect B-B form a atiode there for each wave compoet, which whe superimposed lead to large sigicat wave heights. O top of the shoal (Figure 6) ad alog trasect E-E (Figure 7) where the waves are breakig ad focusig, the same eerg discrepac that was see i Figure 5 is observed, but the spread of the wave heights is simulated reasoabl well b the. The comparisos further behid the shoal (Figures 8? ) o the other had are extremel good. I geeral we see that the wave height distributio behid the shoal is more smoothed out for the broad directioal test cases (Tests 4 ad 6) as compared to the arrow directioal test cases (Tests ad 5). A iterestig observatio is that behid the shoal the wave height distributios are more a fuctio of the tpe of directioal distributio of the iput spectrum, istead of beig a fuctio of the eerg cotet of the spectrum. Before the focusig takes place (Figure 6), the wave height distributios for Tests ad 4, ad Tests 5 ad 6 are quite similar, while after focusig (Figures 6? ) Tests ad 5, ad Tests 4 ad 6 have similar wave height spreadigs. This eect ca also be see clearl i Figure 5, where the wave height distributio till x = 5m is a fuctio of the eerg cotet of the spectrum, ad beod that depeds o the directioal spreadig of the spectrum. Though the gives reasoable sigicat wave height comparisos, it is uable to predict wave-wave iteractios sice it is based o a liear superpositio of moochromatic wave compoets. These iteractios lead to the formatio of higher harmoics i ature, ad become more proouced with icreased oliearit. A compariso of spectra to spectra o top of Chawla, Ozka & Kirb
Sf(cm sec) Sf(cm sec)..4.6.8..4.6.8 f(hz) Iitial spectrum..4.6.8..4.6.8 f(hz) Spectrum o top of the shoal Figure : Frequec spectra comparisos (Test ) showig oliear wave-wave iteractios o top of the shoal the shoal (Figure ) shows this disparit quite clearl. The higher harmoics (secod peak) i the have cosiderable amout of eerg compared to the primar wave eld (rst peak), all of which are ot predicted b the. These higher harmoics are see i the o top of the shoal ad i the regio of focus where the wave eld is highl oliear. CONCLUSIONS A parabolic for simulatig the evolutio of wave spectra over a getl slopig bottom has bee tested with experimetal. Wave height comparisos have show that the works reasoabl well i simulatig wave height distributios for breakig radom waves. Some discrepacies exist i the regio of focus which could be due to the parabolic limitatios of the. Discrepacies could also be due to limitatios of the breakig, ad i predictig o-liear eects. To get a better idea as to whether the discrepac betwee the ad the o top of the shoal is due to the limitatios of the umerical, or errors i the experimetal, comparisos eed to be made to a which will be able to simulate waves with o limitatios o the rage of agles ad also predict the geeratio of higher harmoics o top of the shoal. Noetheless, we d that the spectral works well i simulatig trasformatios of a radom wave eld over a irregular bathmetr, eve for broad directioal spectra. Although certai aspects of the frequec spectrum caot be obtaied accuratel from the, it ca be used to obtai useful estimates of sigicat wave heights. ACKNOWLEDGMENTS This research has bee sposored b the U.S. Arm Corps of Egieers, Coastal Egieerig Research Ceter (Cotract No. DACW 9-9-D-6- D) ad b NOAA Oce of Sea Grat, Departmet of Commerce, uder Grat No. NA/6RG6- (Project No. R/OE-9). The U.S. Govermet is authorized to produce ad distribute reprits for govermetal purposes, ot withstadig a copright otatio that ma appear hereo. Chawla, Ozka & Kirb
REFERENCES Berkho, J. C. W. (97) \Computatio of combied refractio diractio", Proc. th Itl. Cof. Coastal Egrg., ASCE, Vacouver. Booij, N. (98) \Gravit waves o water with o-uiform depth ad curret", Reprt No. 8-, Commuicatio o Hdraulics, Departmet of Civil Egieerig, Delft Uiversit of Techolog. Borgma, L. E. (984) \Directioal spectrum estimatio for the S x gages", Techical Report, Coastal Egrg. Res. Ceter, Vicksburg, -4. Goda, Y. (985) Radom Seas ad Desig of Maritime Structures, Uiv. of Toko Press, Japa. Grassa, J. M. (99) \Directioal radom waves propagatio o beaches", Proc. d Itl. Cof. Coastal Egrg., Delft, 798-8. Isobe, M. (987) \A parabolic for trasformatio of irregular waves due to refractio, diractio ad breakig", Coastal Egieerig i Japa,, -47. Izumia, T. ad Horikawa, K. (987) \O the trasformatio of directioal waves uder combied refractio ad diractio",coastal Egieerig i Japa,, 49-65. Kirb, J. T. (986a) \Higher-order approximatios i the parabolic equatio for water waves", J. Geophs. Res., 9, 9-95. Kirb, J. T. (986b) \Ratioal approximatios i the parabolic equatio method for water waves", Coastal Egieerig,, 55-78. Kirb, J. T. ad Dalrmple, R. A. (98) \A parabolic equatio for combied refractio diractio of Stokes waves b mildl varig topograph", J. Fluid Mech (98), 6, 45-466. Kirb, J. T. ad Dalrmple, R. A. (984) \A vericatio of a parabolic equatio for propagatio of weakl oliear waves", Coastal Egieerig, 8, 9-. Mase, H. ad Kirb J. T. (99) \Modied frequec-domai KdV equatio for radom wave shoalig", Proc. Itl. Cof. Coast. Egrg., Veice. Mei, C. C. (99) The Applied Damics of Ocea Surface Waves, World Scietic, New Jerse O'Reill, W. C. ad Guza, R. T. (99), \Compariso of spectral refractio ad refractio-diractio wave s", J. Waterwa, Port, Coastal ad Ocea Egrg., 7, 99-5. Ozka, H. T. ad Kirb, J. T. (99) \Evolutio of breakig directioal spectral waves i the earshore zoe", Proceedigs of the Secod Iteratioal Smposium, New Orleas, 849-86. Chawla, Ozka & Kirb
Pachag, V. G., Wei, G., Pearce, B. R. ad Briggs, M. J. (99) \Numerical simulatio of irregular wave propagatio over shoal", J. Waterwas, Port, Coastal ad Ocea Egrg., 6, 4-4. Radder, A. C. (979) \O the parabolic equatio method for water wave propagatio", J. Fluid Mech (979), 95, 59-76. Thorto, E. B. ad Guza, R. T. (98) \Trasformatios of wave height distributio", J. Geophs. Res., 88, 595-598. Vicet, C. L. ad Briggs, M. J. (989) \Refractio-diractio of irregular waves over a moud", J. Waterwas, Port, Coastal ad Ocea Egrg., 5, 69-84. 4 Chawla, Ozka & Kirb