NUMERICAL SIMULATION OF INTERNAL WAVES USING A SET OF FULLY NONLINEAR INTERNAL-WAVE EQUATIONS

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1 Aual Joural o Hydraulc Eeer JSCE Vol.5 7 February NUMERICAL SIMULATION OF INTERNAL WAVES USING A SET OF FULLY NONLINEAR INTERNAL-WAVE EQUATIONS Taro KAKINUMA ad Kesuke NAKAYAMA Member o JSCE Dr. o E. Mare Evrome ad Eeer De. Por ad Aror Researc Isue -- Naase Yokosuka Kaaawa 9-86 Jaa) Member o JSCE Dr. o E. Coasal ad Mare De. Naoal Isue or Lad ad Irasrucure Maaeme -- Naase Yokosuka Kaaawa 9-86 Jaa) Ieral waves a wo-layer sysem are smulaed us a se o ully olear eral-wave equaos wc was derved o e bass o a varaoal rcle wou ay assumos o wave oleary ad dsersvy. Comuaoal resuls o erace dslacemes u o eac order o e vercal le scale o moo are comared w calculao resuls obaed us a Boussesq-ye eral-wave model or e es eermeal daa. I a lo-wave case erace dslacemes obaed by e roosed model w more a wo vercally dsrbued ucos o velocy oeal are armoy w ose by e Boussesq-ye model as well as e eermeal daa esecally e wave umber. I a ermedae-wave case e rese model sows dere resuls rom ose rou e Boussesq-ye model wc sould o be aled o s case wou eou cosderao o e wave dsersvy. Key Words : Ieral wave ully olear equao varaoal rcle wo-layer model umercal smulao. INTRODUCTION I a lake or e ocea were desy sracao s well develoed o oly eral loerod waves e.. eral seces ad des bu also eral sor-erod waves ca be observed; ) e sources o e laer clude boom ooray ad eracal sably. Suc eral waves ecae er eery amo comoes over a wde wave-requecy bad w sro oleary esecally we ey reac sallow waer reos. Te oleary ad dsersvy o eral waves owever ave bee closely suded or oly eral lo waves were eral-lo-wave equaos were derved o evaluae erace roles wou dsurbace or m due o umercal errors. For eamle o a erurbao bass Co ad Camassa ) derved wo ses o eral-wave equaos cosder ull oleary o eral waves a wo-layer sysem were we coose oe se o equaos o rea a sallow layer weer les o aoer sallow layer w weak dsersvy or a dee layer w ermedae dsersvy. O e oer ad Kakuma ) dd o use ay assumo o oleary as well as dsersvy o eral waves e dervao rocess o ully olear eral-wave equaos based o a varaoal rcle. For s reaso e alcao o s model s eorecally ree rom lmaos cocer e relave ckess o lud layers or e requecy bad o ully olear ad ully dsersve surace/eral waves. I e rese aer s se o equaos as bee solved umercally o smulae eral waves a wo-layer sysem u o eac order o e vercal le scale o erace dslaceme. Comuaoal resuls are comared w e es eermeal daa 4) or calculao resuls rou a se o Boussesq-ye eral-wave equaos. 5). FULLY NONLINEAR EQUATIONS FOR SURFACE/INTERNAL WAVES ) Mullayer luds Ivscd ad comressble luds are assumed o be sable sll waer as sow F. were

2 ese luds are rereseed as I) rom o o boom. Te -layer ckess sll waer s deoed by ). Noe o e luds m eve w moo. Te desy s saally uorm ad emorally cosa eac layer were < < < I. Surace eso ad callary aco are eleced. Flud moo s assumed o be rroaoal resul e esece o velocy oeal φ deed as u ad w φ / z ) φ were s a aral dereal oeraor e orzoal lae.e. / / y). ) Fucoal or e varaoal roblem Te ressure o z.e. e lower erace o e -layer s wre by ). I e -layer bo e elevao o oe erace z - ) or ) ad e ressure o e oer erace - ) are kow e e ukow varables are e velocy oeal φ z ) ad erace elevao ) suc a e ucoal or e varaoal roblem e -layer F s deermed by F φ [ φ ] φ ) ) were P ) k k ; s ravaoal accelk erao; e lae A wc s e orooal roeco o e obec doma oo e -y lae s assumed o be deede o me. I comarso w e ucoal reerred o Luke 6) or roaoal moo Eq. ) as a addoal erm o e eracal ressure wou e erms rela o vorcy. ) Vercally dsrbued ucos I order o derve a se o equaos wose ye s orzoally wo-dmesoal vercal erao s erormed aalycally. I a maer smlar o a or e ully olear surace-wave model 7) e velocy oeal φ s eaded o a seres erms o a ve se o vercally dsrbued ucos mulled by er wes.e. N z ) z ) ) ) φ ) A P z dz da d φ z were N s e umber o vercally dsrbued ucos ad e sum rule o roduc s adoed or subscr. F. Mullayer lud sysem. 4) Euler-Larae equaos uder varaoal rcle We subsue Eq. ) o Eq. ) aer wc e ucoal F s eraed vercally. Te e varaoal rcle s aled o oba e ollow Euler-Larae equaos.e. e ully olear equaos or surace ad eral waves: ) dz z z 4) 5) e were N; e z e or ); / z z. 5) Two-layer model bewee orzoal laes We cosder wo-layer roblems bewee wo ed orzoal laes were ad ) D. Te erace role s descrbed by z ) were. I s aer e vercally dsrbued uco s deermed by z k /. 6) k [-layer] I e -layer.e. e uer layer ad. We roduce deed as z 7) aer wc Eq. 6) s subsued o Eqs. 4) ad 5) resul / z z dz P { ) } ) 8)

3 9) [-layer] I e -layer.e. e lower layer ad. We roduce deed as ) aer wc Eq. 6) s subsued o Eqs. 4) ad 5) lead o ) ). NUMERICAL CALCULATION METHOD Te ully olear wo-layer equaos.e. Eqs. 8) 9) ) ad ) are rewre o e derece equaos as ) 4) 5) a) Dmesos o e laboraory ak. 4) b) Ial codo o calculao. Te erace s cled learly w e ale θ e orzoal ak. F. Scemac o aks. 6) resecvely. I sould be oed a e su dcaes e rd-o umber Eqs. ) 6). Te me develome s carred ou by aly mlc scemes. 4. NUMERICAL SIMULATION OF LONG WAVES Two-layer roblems are solved vercally wo- dmesoal cases. Hor e al. 4) erormed ydraulc eermes us a ak wose le L de D ad wd W were 6. m.9 m ad. m resec- vely as sow F. a). Tree ulra-soc wave aues were se a e osos marked A B ad C. Ts ak was lled w a wo-layer sra- cao were /D.8 aer wc was roaed very slowly rou θ aroud e as o roao. A e be o e eermes s led ak was reured o a orzoal oso very quckly. I e al codo o umercal comuaos e ak s orzoal ad e erace s cled learly as sow F. b). Te rd wd ad me-se erval are equal o.6 m ad. s resecvely rouou every comuao erormed e rese sudy. I Fs. 5 e eermeal ad calculao. [ ] { } { } ) ). { } { } ) )

4 F. Tme seres o erace dslaceme measured by e wave aue a oso C e ydraulc eerme. F. 4 Tme seres o erace dslaceme obaed by e Boussesq-ye model corresod o a F.. resuls are comared or e me seres o erace dslacemes a e oso marked C F. a) e case were e desy rao / s.9 ad e l ale θ s.467. F. sows e eermeal resul measured by e wave aue a oso C. F. 4 sows e corresod calculao resul rou e Boussesq-ye model 5) BT) wose udameal equaos are wre APPENDIX. F. 5 sows e calculao resuls obaed us e roosed ully olear model FN) clud e cases were e umber o vercally dsrbued ucos or velocy oeal N s equal o 4 ad 5. We N e se o ully olear eralwave equaos.e. Eqs. 8) 9) ) ad ) reduces o a se o olear ad o-dsersve eral-wave equaos wc sows ereme dserao aroud e wave cress as sow F. 5 wou dsersvy balac w oleary. We N e FN akes o accou lear ad uorm dsrbuos o u ad w e dreco o z resecvely suc a e balace bewee e oleary ad dsersvy s cosdered lead o e more accurae resul a a we N. We N e erace dslaceme evaluaed by e FN s closer o a F. 4 obaed by e BT were e arabolc dsrbuo o u e dreco o z s cosdered bo e FN ad BT. I sould be oed a ou e eec due o e lear dsrbuo o w e dreco o z s F. 5 Tme seres o erace dslacemes obaed by e rese ully olear model corresod o a F.. Te resuls are sow or dere umbers o vercally dsrbued ucos velocy oeal N. F. 6 Ierace roles obaed by e rese ully olear model were N 4 ad e Boussesq-ye model we 8 s. cosdered bo e FN ad BT e FN esmaes e wave erod or e wave umber more accuraely by solv e corbuo o eac order wou erurbao wle e wave erod rou e BT s loer a a o e eermeal daa. Te erace dslacemes obaed by e FN

5 F. 7 Tme seres o erace dslaceme obaed by e Boussesq-ye model or e ermedae-waves. ardly sow derece bewee e cases were N 4 ad 5. Alou bo e FN ad BT do o clude dssao eecs due o rco wc resuls o larer wave es a e eermeal daa e armoy o resuls bewee e FN ad BT dcaes e accuracy o resuls calculaed by e FN s lo-wave codo. Te erace roles obaed us e FN were N 4 ad e BT we 8 s are sow F. 6 accord o wc e rereseave rao o waer de o wavele /λ s abou.6. Te FN esmaes e wave es larer ad e waveles sorer w er oleary a e BT. 5. NUMERICAL SIMULATION OF INTERMEDIATE WAVES I s seco a deeer case were e rereseave rao /λ s aroud. s reaed umercal comuaos. Te le L ad de D o e ak sow F. b) are. m ad.5 m resecvely. Te desy rao / s. wle e sll waer de rao /D s.8 also s case. I e al codo o calculao e l ale θ s equal o 5.. Fs. 7 ad 8 sow e me seres o erace dslacemes a e ceer o e ak e ormer o wc was obaed us e BT wle e laer was evaluaed by e FN we N or 4. I s ermedae-wave case e resuls sow muc derece bewee e BT ad FN wc suess a e BT s o alcable because e se o Boussesq-ye equaos was derved o e bass o a erurbao aroud e lo-wave codo w oly weak dsersvy as well as weak oleary. We N see F. 8 e eral waves o e FN sow orward leas o er wave roles wc are resraed we N s larer a oe. I sould be oed a eve we N or e FN evaluaes e eral-wave erods sorer a F. 8 Tme seres o erace dslacemes obaed by e rese ully olear model corresod o a F. 7. Te resuls are sow or dere umbers o vercally dsrbued ucos velocy oeal N. F. 9 Vercal dsrbuos o orzoal velocy e lower layer u were.6 m we s. Te resul was obaed us e FN we N 4. Eac le sows e velocy dsrbuo were e comoes are added oeer u o or o.

6 ose calculaed by e BT wc esmaes e wave erod loer a a o e acual ermedae waves as e eral waves were lo waves. F. 9 sows e vercal dsrbuos o orzoal velocy u e lower layer a a eralwave cres were.6 m we s. Te resul was obaed us e FN we N 4 ad draw e ure or eac case were e comoes u o or o are added oeer. Alou a e eral-wave cres e derece bewee u s o e erace ad boom s abou 5. % o u o e boom e derece could be mora o evaluae eral-wave roles accuraely cosder e dsersvy o ermedae waves. 6. CONCLUSIONS Te eral waves a wo-layer sysem ave bee smulaed us e se o ully olear eral-wave equaos. Te comuaoal resuls o erace dslacemes u o eac order o e vercal le scale o moo were comared w e corresod calculao resuls obaed us e Boussesq-ye eral-wave model or e es eermeal daa. I e lo-wave case e erace dslacemes esmaed by e roosed model w more a ree vercally dsrbued ucos o velocy oeal were armoy w a rou e Boussesq-ye model as well as e eerme esecally e wave umber. I e ermedae-wave case e roosed model sowed dere resuls rom ose by e Boussesq-ye model wc sould o be aled o s case wou eou cosderao o wave dsersvy. I comuaos us e roosed ully olear model w eou umber o vercally dsrbued ucos o velocy oeal e wes o comoes are evaluaed based o e varaoal rcle wou assumos o oleary ad dsersvy o waves suc a s model s eeced o be aled o mullayer sysems clud waves o varous requeces over ooraes o vesae or eamle eerao mecasms o eral sor-erod waves rom lo-erod waves suc as surace/eral des. APPENDIX BOUSSINESQ-TYPE EQUATIONS FOR INTERNAL WAVES We also ulze e eral-lo-wave model 5) wose udameal equaos are e Boussesqye equaos or olear eral waves.e. [-layer] [-layer] φ φ { ) φ } φ ) φ φ ) { ) φ } ) 7) 8) 9) ) were φ ad φ are velocy oeals e uer ad lower layers resecvely. Te umercal scemes w e derece meods o solve ese equaos are resemble o ose aled e roosed ully olear model. REFERENCES ) Robers J.: Ieral Gravy Waves e Ocea Mare Scece ed. Hood D. W.) Marcel Dekker Ic. Vol ) Co W. ad Camassa R.: Fully olear eral waves a wo-lud sysem J. Flud Mec. Vol ) Kakuma T.: A se o ully olear equaos or surace ad eral ravy waves Proc. o e 5 I. Co. o Comuer Modell o Seas ad Coasal Reos WIT Press ) Hor D. A. Redeko L. G. Imberer J. ad Ivey G. N.: Ieral wave evoluo a sace-me vary eld J. Flud Mec. Vol ) Nakayama K. ad Imberer J.: Resdual crculao due o eral waves soal o a sloe J. Pys. Oceaor. revso) 6) Luke J. C.: A varaoal rcle or a lud w a ree surace J. Flud Mec. Vol ) Isobe M.: Tme-deede mld-sloe equaos or radom waves Proc. o e 4 I. Co. o Coasal E. ASCE φ Receved Seember 6)